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University  of  California  •  Berkeley 

The  Theodore  P.  Hill  Collection 

of 
Early  American  Mathematics  Books 


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ORTON'S 

LIGHTNING  CALCULATOR, 

AND 

THE   SHORTEST,   SIMPLEST,    AND   MOST  RAPID   METHOD  OF  COMPUTING 

>'IJMBP:RS,     ADAPTED    TO     EVERT     KIND     OF^PSINESS,     AND 

WITHIN   THE   COMPREHENSION   OF   EVERT   ONE   HAVING 

THE    SLIGHTEST    KNOWLEDGE    OF    FIG0RB3. 

BY 

HOY  D.   ORTON. 

ENTIRELY  WE^S^V  EDITIOW, 

WITH    EXTEKSIVE    MODIFICATIONS   AND    IMPROVEMENTS. 


N.  B.— Any  Infl-lncement  upon  the  eopyrlsht  of  this  book  will 
prosecuted  to  the  fullest  extent  of  the  law. 


PHILADELPHIA: 
COLLINS,    PRINTER, 

705  JAYNE  STREET. 


Entered  according  to  Act  of  Congress,  in  the  year  1871,  by 

HOY  D.  ORTON, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


N.  B. — It  gives  me  pleasure  to  state  that,  in  the  revision  of 
this  book,  I  have  been  deeply  indebted  to  S.  J.  Donaldson, 
Jr.,  of  Baltimore,  a  gentleman  favorably  known  as  the  author 
of  "Lyrics,  and  Other  Poems." 


COLLINS,    PRINTER. 


INTRODUCTION. 

Quantity  is  that  which  can  be  increased  or 
diminished  by  augments  or  abatements  of  homo- 
geneous parts.  Quantities  are  of  two  essential 
kinds,  Geometrical  and  Physical, 

1.  Geometrical  quantities  are  those  which  occupy 
space ;  as  Unes^  surfaces^  solids,  liquids,  gases,  etc. 

2.  Physical  quantities  are  those  which  exist  in 
the  time,  but  occupy  no  space ;  they  are  known  by 
their  character  and  action  upon  geometrical  quan- 
tities, as  attraction,  light,  heat,  electricity  and  mag^ 
netism,  colors,  force,  power,  etc. 

To  obtain  the  magnitude  of  a  quantity  we  com- 
pare it  with  a  part  of  the  same ;  this  part  is  im- 
printed m  our  mind  as  a  unit,  by  which  the  whole 
is  measured  and  conceived.  No  quantity  can  be 
measured  by  a  quantity  of  another  kind,  but  any 
quantity  can  be  compared  with  any  other  quantity, 
and  by  such  comparison  arises  what  we  call  calcu' 
lation  or  Mathematics. 


lY  INTRODUCTION. 

MATHEMATICS. 
Mathematics  is  a  science  by  which  the  com- 
parative value  of  quantities  are  investigated ;  it  is 
divided  into : 

1.  Arithmetic,  that  branch  of  Mathematics 
which  treats  of  the  nature  and  property  of  num- 
bers ;  it  is  subdivided  into  Addition^  Subtraction^ 
Multiplication^  Division^  Involution^  Evolution  and 
Logarithms. 

2.  Algebra,  that  branch  of  Mathematics  which 
employs  letters  to  represent  quantities,  and  by  that 
means  performs  solutions  without  knowing  or 
noticing  the  value  of  the  quantities.  The  subdi- 
visions of  Algebra  are  the  same  as  in  Arithmetic. 

3.  Geometry,  that  branch  of  Mathematics  which 
investigates  the  relative  property  of  quantities  that 
occupies  space;  its  subdivisions  are  Longemetry^ 
Planemetry^  Stereometry^  Trigonometry  and  Conic 
Sections. 

4.  Differential-calculs,  that  branch  of  Math- 
ematics which  ascertains  the  mean  effect  produced 
by  group  of  continued  variable  causes. 

5.  Integral-calculs,  the  contrary  of  Differen- 
tial, or  that  branch  of  Mathematics  which  investi- 
gates the  nature  of  a  continued  variable  cause  that 
has  produced  a  known  effect. 


PREFACE 


Mathematical  laws  are  tlie  acknowledged 
basis  of  all  science.  Ever  since  the  streets  of 
Athens  resounded  with  that  historical  cry  of  "Eu- 
reka," emanating  from  one  of  antiquity's  greatest 
mathematicians,  the  science  has  been  steadily  pro- 
gressing. 

It  is  not  our  purpose,  in  this  small  work,  to  in- 
troduce any  of  the  higher  branches  of  mathematics, 
VIZ.:  Algebra,  Conic  Sections,  Calculus,  etc.  Our 
object  is  merely  to  present  to  the  public  a  system 
of  calculation  that  is  practical  to  every  business 
man.  It  consists  of  the  addition  of  numbers  on  a 
principle  entirely  different  from  the  one  ordinarily 
used.  In  the  practical  application  of  this  new  prin- 
ciple of  addition,  scarcely  any  mental  labor  is  re- 
quired, compared  with  the  principle  of  addition  set 
forth  in  standard  works.  The  superiority  we  claim 
for  this  principle  above  all  others,  is  this,  that  it 
requires  no   great  mental  exertion,  affording  the 


Tl  PREFACE. 

preatest  facilities  to  tbe  calculator  in  the  addition 
of  numbers,  enabling  him  to  add  a  whole  day  with- 
out any  mental  fatigue ;  whereas,  by  the  ordinary 
way,  it  is  very  laborious  and  fatiguing. 

Our  system  of  calculation  also  embraces  a  concise, 
rapid,  and  at  the  same  time  practical  method  of 
Multiplication,  by  which  one  is  enabled  to  arrive 
ftt  the  product  of  any  number  of  figures,  multiplied 
by  any  number,  immediately,  without  the  use  of 
partial  products. 

This  small  work  also  embraces  the  shorteyt  and 
most  concise  method  for  the  computation  of  Interest 
ever  introduced  to  the  public.  Our  system  for  com- 
puting interest  is  entirely  diflferent  from  any  rule 
ever  introduced,  for  the  computation  of  either  Sim- 
ple or  Compound  Interest.  A  student  having  gone 
no  further  than  Long  Division  in  Arithmetic,  can, 
by  our  rule,  calculate  Simple  or  Compound  Interest 
at  any  given  rate  per  cent.,  for  any  given  time,  in 
one-tenth  of  the  time  that  the  best  calculators  will 
compute  it  by  the  rules  laid  down  in  other  books. 
By  using  our  rules,  you  can  entirely  avoid  the  use 
of  fractions,  and  save  the  calculation  of  75  to  100 
figures,  where  years,  months  and  days  are  given  oa 
a  note. 


ADDITION. 


N  B.  The  above  process  of  addition  is  only  re- 
commended  for  beginners. 

Process. — For  adding  the  above  example,  com- 
mence at  the  bottom  of  the  right-hand  column. 
Add  thus ;  12,  16,  22 ;  then  carry  the  2  tens  to 
the  second  column,  then  add  thus,  8,  10,  18,  22^ 
carry  the  two  hundreds  to  the  third  column,  and 
add  the  same  way,  9,  13,  16,  23.  Never  permit 
yourself  for  once  to  add  up  a  column  in  this  man- 
ner, 3  and  9  are  12  and  4  are  16,  and  6  are  22  ;  it 
is  just  as  easy  to  name  the  sum  at  once,  without 
naming  the  figures  you  add,  and  three  times  as 
rapid, 

9 


10         orton's  lightning  calculator, 
addition  of  short  columns  of  figures. 

Addition  is  the  basis  of  all  numerical  opera- 
tions, and  is  used  in  all  departments  of  business, 
To  aid  the  business  man  in  acquiring  facility  and 
accuracy  in  adding  short  columns  of  figures,  the 
following  method  is  presented  as  the  best : 

Process. — Commence  at  the  bottom  of 
274 
oj^g     the  right-hand  column,  add  thus:  16,  22, 

134     ^2 ;  then  carry  the  3   tens  to   the  second 

342     column ;  then  add  thus :  7,  14,  25 ;  carry 

^27     the  2  hundreds  to  the  third  column,  and 

^^^     add   the   same  way:  12,  16,  21.     In  this 

2152     'vvay  you  name  tli3  sum  of  two  figures  at 

once,  which  is  quite  as  easy  as  it  is  to  add  one 

figure  at  a  time.     Never  permit  yourself /or  once 

to  add  up  a  column  in  this  manner :  9  and  7  are 

16,  and  2  are  18   and  4  are  22,  and  6  are  28,  and 

4  are  32.     It  is  just  as  easy  to  name  the  result 

of  two  figures  at  once  and  four  times  as  rapid. 

The  following  method  is  recommended  for  the 

addition  OF  LONG  COLUMNS  OF  FIGURES. 

In  the  addition  of  long  columns  of  figures 
which  frequently  occur  in  books  of  accounts,  in 
order  to  add  them  with  certainty,  and,  at  the 
same  time,  with  ease  and  expedition,  study  well 
the  following  method,  which  practice  will  render 
familiar,  easy,  rapid,  and  certain. 


ADDITION.  11 

THE    EASY    WAY    TO    ADD. 

EXAMPLE  2— EXPLANATION. 

Commence  at  9  to  add,  and  add  as  near  20  as  pos- 
sible, thus:  94-2-1-4+3=18,  place  the  8  to  the 
right  of  the  3,  as  in  example ;  commence  at  6  to  7' 
add  6-f  4-i-8=18  ;  place  the  8  to  the  right  of  4 
the  8,  as  in  example  j  commence  at  6  to  add  G 
6+4-f7=17  ;  place  the  7  to  the  right  of  the  3« 
7.  as  in  example ;  commence  at  4  to  add  4-|-  9 
9-f  3=16  ;  place  the  6  to  the  right  of  the  3,  4 
as  in  example;  commence  at  6  to  add  6-f4  7' 
-f-7=17 ;  place  the  7  to  the  right  of  the  7,  4 
as  in  example;  now,  having  arrived  at  the  6 
top  of  the  column,  we  add  the  figures  in  the  8** 
new  column,  thus:  7-}-6-|-7-f8-J-8=36;  place  4 
the  right  hand  figure  of  36,  which  is  a  6,  6 
under  the  original  column,  as  in  example,  and  3' 
add  the  left  hand  figure,  which  is  a  3,  to  the  4 
number  of  figures  in  the  new  column;  there  2 
are  5  figures  in  the  new  column,  therefore  9 
3-[-5=8 ;  prefix  the  8  with  the  6,  under  the  — 
original  column,  as  in  example ;  this  makes  86 
86,  which  is  the  sum  of  the  column. 

Remark  1. — If,  upon  arriving  at  the  top  of  the 
column,  there  should  be  one,  two  or  three  figures 
whose  sum  will  not  equal  10,  add  them  on  to  the 
Bum  of  the  figures  of  the  new  culumn,  never  plyciug 


12  orton's  lightning  calculator. 

an  extra  figure  in  the  new  column,  unless  it  be  an 
excess  of  units  over  ten. 

Remarh  2. — By  this  system  of  addition  you  can 
stop  any  place  in  the  column,  where  the  sum  of  the 
figures  will  equal  10  or  the  excess  of  10 ;  but  the 
addition  will  be  more  rapid  by  your  adding  as 
near  20  as  possible,  because  you  will  save  the  form- 
ing of  extra  figures  in  your  new  column. 
EXAMPLE— EXPLANATION. 

2+6+7=15,  drop  10,  place  the  5  to  the  right 
of  the  7;  6+5+4=15,  drop  10,  place  the  5  to 
the  right  of  the  4,  as  in  example;  8+3+7=18, 
drop  10,  place  the  8  to  the  right  of  the  7,  4 
as  in  example ;  now  we  have  an  extra  figure,  7^ 
which  is  4  ;  add  this  4  to  the  top  figure  of  the  3 
new  column,  and  this  sum  on  the  balance  of  8 
the  figures  in  the  new  column,  thus:  4+8+  4* 
5+5=22  ;  place  the  right  hand  figure  of  22  5 
under  the  original  column,  as  in  example,  and  6 
add  the  left  hand  figure  of  22  to  the  num-  7' 
ber  of  figures  in  the  new  column,  which  are  6 
three,  thus :  2+3=5 ;  prefix  this  5  to  the  2 
figure  2,  under  the  original  column ;  this  — 
makes  52,  which  is  the  sum  of  the  cokmn.         52 


ADDITION.  13 

"Rule.-— 2^ or  adding  two  or  more  coIumnSy  com- 
mence at  the  right  handj  or  units*  column;  proceed 
in  the  sam^e  manner  as  in  adding  one  column;  after 
the  sum  of  the  first  column  is  obtained,  add  all 
except  the  right  hand  figure  of  this  sum  to  the  second 
column,  adding  the  second  column  the  same  wag  you 
added  the  first;  proceed  in  like  manner  with  all  the 
columns,  always  adding  to  each  successive  column 
the  sum  of  the  column  in  the  next  lower  order,  minus 
ilie  right  hand  figure. 

N.  B.  The  small  figures  wliicli  we  place  to  tlie 
riglit  of  the  column  when  adding  are  called  integers. 

The  addition  by  integers  or  by  forming  a  new 
column,  as  explained  in  the  preceding  examplea 
should  be  used  only  in  adding  very  long  columns 
of  figuies,  say  a  long  ledger  column,  where  the  foot- 
ings of  each  column  would  be  two  or  three  hundred, 
in  which  case  it  is  superior  and  much  more  easy 
than  any  other  mode  of  addition ;  but  in  adding 
short  columns  it  would  be  useless  to  form  an  extra 
column,  where  there  is  only,  say,  six  or  eight  fig- 
ures to  be  added.  In  making  short  additions,  the 
following  suggestions  will,  we  trust,  be  of  use  to 
the  accountant  who  seeks  for  information  on  this 
subject. 

In  the  addition  of  several  3olumns  of  figures, 
where  they  are  only  four  or  five  deep,  or  when 
their  respective  sums  will  range  from  twenty  ^ve 


'14  orton's  lightning  calculator. 

to  forty,  the  accountant  should  commence  with  the 
unit  column,  adding  the  sum  of  the  first  two  figures 
to  the  sum  of  the  next  two,  and  so  on,  naming  only 
the  results,  that  is,  the  sum  of  every  two  figures. 

In  the  present  example  in  adding  the  unit  346 
column  instead  of  saying  8  and  4  are  12  and  235 
5  are  It  and  6  are  23,  it  is  better  to  let  the  T24 
eye  glide  up  the  column  reading  only,  8,  12,  598 
It,  23;  and  still  better,  instead  of  making  a 
separate  addition  for  each  figure,  group  the  figures 
thus:  12  and  11  are  23,  and  proceed  in  like  man- 
ner with  each  column.  For  short  columns  this  is 
a  very  expeditious  way,  and  indeed  to  be  preferred ; 
but  for  long  columns,  the  addition  by  integers  is 
the  most  useful,  as  the  mind  is  relieved  at  intervals 
and  the  mental  labor  of  retaining  the  whole  amount, 
as  you  add,  is  avoided,  which  is  very  important  to 
any  person  whose  mind  is  constantly  employed  in 
various  commercial  calculations. 

In  adding  a  long  column,  where  the  figures  are 
of  a  medium  size,  that  is,  as  many  8s  and  9s  as 
there  are  2s  and  3s,  it  is  better  to  add  about  three 
figures  at  a  time,  because  the  eye  will  distinctly  see 
that  many  at  once,  and  the  ingenious  student  will 
in  a  short  time,  if  he  adds  by  integers,  be  able  to 
read  the  amount  of  three  figures  at  a  glance,  or  as 
quick,  we  might  say,  as  he  would  read  a  single 
figure. 


ADDITION.  15 

Here  we  begin  to  add  at  the  bottom  of  the     *26« 
unit  column  and  add  successively  three  fig-       "• 
ures  at  a  time,  and  place  their  respective       004 
sums,  minus  10,  to  the  right  of  the  last  fig-     95^ 
are  added;  if  the  three  figures  do  not  make       62 
10,  add  on  more  figures;  if  the  three  figures       87^ 
make  20  or  more,  only  add  two  of  the  fig-     ^^^ 
ures.     The  little  figures  that  are  placed  to       ^^ 4 
the  right  and  left  of  the  column  are  called     877 
integers.     The  integers  in  the  present  ex-       33 
ample,  belonging  to  the  units  column,  are       84* 
4,  4,  5, 4,  6,  which  we  add  together,  making       ^^ 

23;  place  down  3  and  add  2  to  the  number  

of  integers,  which  gives  7,  which  we  add  to     803 
the  tens  and  proceed  as  before. 

Reason. — In  the  above  example,  every  time  wo 
placed  down  an  integer  we  discarded  a  ten,  and 
when  we  set  down  the  3  in  the  answer  we  dis- 
carded two  tens;  hence,  we  add  2  on  to  the  num- 
ber of  integers  to  ascertain  how  many  tens  were 
discarded;  there  being  5  integers  it  made  7  tens, 
which  we  now  add  to  the  column  of  tens;  on  the 
same  principle  we  might  add  between  20  and  30, 
always  setting  down  a  figure  before  we  got  to  30; 
then  every  integer  set  down  would  count  for  2  tens, 
being  discarded  in  the  same  way,  it  does  in  the 
present  instance  for  one  ten.  When  we  add  be- 
tween  10  and  20,   and  in  very  long  columns,  it 


16         orton's  ligutninq  calculator. 

would  be  much  better  to  go  as  near  30  as  possible, 
and  count  2  tens  for  every  integer  set  down,  in 
which  case  we  would  set  down  about  one-half  as 
many  integers  as  when  we  write  an  integer  for 
every  ten  we  discard. 

When  adding  long  columns  in  a  ledger  or  day- 
book, and  where  the  accountant  wishes  to  avoid  the 
writing  of  extra  figures  in  the  book,  he  can  place  a 
strip  of  paper  alongside  of  the  column  he  wishes 
to  add,  and  write  the  integers  on  the  paper,  and  in 
this  way  the  column  can  be  added  as  convenient 
almost  as  if  the  integers  were  written  in  the  book. 

Perhaps,  too,  this  would  be  as  proper  a  time  as 
any  other  to  urge  the  importance  of  another  good 
habit;  I  mean  that  of  making  ylain  figures.  Some 
persons  accustom  themselves  to  making  mere 
scrawls,  and  important  blunders  are  often  the  result. 
If  letters  be  badly  made  you  may  judge  from 
such  as  are  known;  but  if  one  figure  be  illegible, 
its  value  can  not  be  inferred  from  the  others.  The 
vexation  of  the  man  who  wrote  for  2  or  3  monkeys, 
and  had  203  sent  him,  was  of  far  less  importance 
than  errors  and  disappointments  sometimes  result- 
ing from  this  inexcusable  practice. 

We  will  now  proceed  to  give  some  methods  of 
proof.  Many  persons  are  fond  of  proving  the  cor- 
rectness of  work,  and  pupils  are  often  instructed 
to  do  so,  for  the  double  purpose  of  giving  them 


ADDITION.  17 

exercise  in  calculation  and  saving  their  teacher  the 
trouble  of  reviewing  their  work. 

There  are  special  modes  of  proof  of  elementary 
operations,  as  by  casting  out  threes  or  nines,  or  by 
changing  the  order  of  the  operation,  as  in  add- 
ing upward  and  then  downward.  In  Addition, 
some  prefer  reviewing  the  work  by  performing  the 
Addition  downward,  rather  than  repeating  the 
ordinary  operation.  This  is  better,  for  if  a  mis- 
take be  inadvertently  made  in  any  calculation, 
and  the  same  routine  be  again  followed,  we  are  very 
liable  to  fall  again  into  the  same  error.  If,  for 
instance,  in  running  up  a  column  of  Addition  you 
should  say  84  and  8  are  93,  you  would  be  liable,  ia 
going  over  the  same  again,  in  the  same  way  to 
slide  insensibly  into  a  similar  error;  but  by  begin- 
ning at  a  different  point  this  is  avoided. 

This  fact  is  one  of  the  strongest  objections  to 
the  plan  of  cutting  off  the  upper  line  and  adding 
it  to  the  sum  of  the  rest,  and  hence  some  cut  off 
the  lower  line  by  which  the  spell  is  broken.  The 
most  thoughtless  can  not  fail  to  see  that  adding  a 
line  to  the  sum  of  the  rest,  is  the  same  as  adding  it 
in  with  the  rest. 

The  mode  off  proof  by  casting  out  the  nines 
and  threes  will  be  fully  explained  in  a  following 
chapter. 

A  very  excellent  mode  of  avoiding  error  in  add- 


18         okton's  ligctnino  calculator. 

ing  long  columns  is  to  set  down  tlie  result  of  each 
column  on  some  waste  spot,  observing  to  place  the 
numbers  successively  a  place  further  to  the  left 
each  time,  as  in  putting  down  the  product  figures 
in  multiplication;  and  afterward  add  up  the 
amount.  In  this  way  if  the  operator  lose  his 
count,  he  is  not  compelled  to  go  back  to  units,  but 
only  to  the  foot  of  the  column  on  which  he  is  op- 
erating. It  is  also  true  that  the  brisk  accountant, 
who  thinks  on  what  he  is  doing,  is  less  liable  to 
err  than  the  dilatory  one  who  allows  his  mind  to 
wander.  Practice  too  will  enable  a  person  to  read 
amounts  without  naming  each  figure,  thus  instead 
of  saying  8  and  6  are  14,  and  7  are  21  and  5  are  2G, 
it  is  better  to  let  the  eye  glide  up  the  column,  read- 
ing only  8,  14,  21,  26,  etc.;  and,  still  further,  it  is 
quite  practicable  to  accustom  one's  self  to  group  87 
the  figures  in  adding,  and  thus  proceed  very  rap-  23 
idly.  Thus  in  adding  the  units'  column,  instead  45 
of  adding  a  figure  at  a  time,  we  see  at  a  glance  62 
that  4  and  2  are  6,  and  that  5  and  3  are  8,  then  24 
6  and  8  are  14;  we  may  then,  if  expert,  add  — 
constantly  the  sum  of  two  or  three  figures  at  a  time, 
and  with  practice  this  will  be  found  highly  advan- 
tageous in  long  columns  of  figures;  or  two  or  three 
columns  may  be  added  at  a  time,  as  the  practiced 
eye  will  see  that  24  and  62  are  86  almost  as  readily 
as  that  4  and  2  are  6. 


ADDITION.  19 

Teachers  will  find  the  following  mode  of  matx3h- 
ing  lines  for  beginners  very  convenient,  as  they  can 
inspect  them  at  a  glance : 

Add  7G54384 
8786286 
3408698 
2345615 
1213713 


23408696 


In  placing  the  above  the  lines  are  matched  in 
pairs,  the  digits  constantly  making  9.  In  the 
above,  the  first  and  fourth,  second  and  fifth  are 
matched;  and  the  middle  is  the  hey  line^  the  result 
being  just  like  it,  except  the  units'  place,  which  is 
as  many  less  than  the  units  in  the  key  line  as  there 
are  pairs  of  lines;  and  a  similar  number  will  oc- 
cupy the  extreme  left.  Though  sometimes  used  aa 
a  puzzle,  it  is  chiefly  useful  in  teaching  learners ; 
and  as  the  location  of  the  key  line  may  be  changed 
in  each  successive  example,  if  necessary,  the  arti« 
fice  could  not  be  detected.  The  number  of  lines 
is  necessarily  odd. 


SHORT  METHODS  OF  MULTIPLICATION. 


Rule. — Set  down  the  smaller  factor  under  the 
larger,  units  under  units,  tens  under  tens.  Begin 
with  the  unit  figure  of  the  multiplier,  multiply  by  it, 
first  the  units  of  the  multiplicand,  setting  the  units 
of  the  product,  and  reserviny  the  tens  to  be  added 
to  the  next  product;  now  multiply  the  tens  of  the 
multiplicand  by  the  unit  figure  of  the  multijMer, 
and  the  units  of  the  multiplicand  by  tens  figure  of 
20 


MULTIPLICATION.  21 

the  multiplier ;  add  these  two  products  together^  set- 
ting down  the  units  of  their  sum,  and  reserving  the 
tens  to  he  added  to  the  next  'product ;  now  multiply 
the  tens  of  the  multiplicand  hy  the  tens  figure  of  the 
multiplier,  and  set  down  the  whole  amount.  This 
will  he  the  complete  product. 

Remark. — Always  add  in  the  tens  that  are  re- 
served as  soon  as  you  form  the  first  produeb. 

EXAMPLE  1.— EXPLANATION. 

1.  Multiply  the  units  of  the  multiplicand       24 
by  the  unit  figure  of  the  multiplier,  thus:       31 

1X4  is  4 ;  set  the  4  down  as  in  example.     

2.  Multiply  the  tens  in  the  multiplicand  by  744 
the  unit  figure  in  the  multiplier,  and  the  units  in 
the  multiplicand  by  the  tens  figure  in  the  multi- 
plier, thus :  1X2  is  2;  3X4  are  12,  add  these  two 
products  together,  2-|-12  are  14,  set  the  4  down 
as  in  example,  and  reserve  the  1  to  be  added  to  the 
next  product.  3.  Multiply  the  tens  in  the  multi- 
plicand by  the  tens  figure  in  the  multiplier,  and 
add  in  the  tens  that  were  reserved,  thus :  3x2  are 
6,  and  6-f  1=7 ;  now  set  down  the  whole 
amount,  which  is  7. 

EXAMPLE  2.— EXPLANATION. 

1.  Multiply  units  by  units,  thus:  4x3         53 
are  12,  set  down  the  2  and  reserve  the  1  to         84 

carry.    2.  Multiply  tens  by  units,  and  units     

by  tens,  and  add  in  the  one  to  carry  on  the     4152 


22  ORTON  &  fullee's  arithmetic. 

first  product,  then  add  these  two  products  together, 
thus:  4X5  are  20+1  are  21,  and  8X3  are  24, 
and  21-f  24  are  45,  set  down  the  5  and  reserve 
the  4  to  carry  to  the  next  product.  3.  Multiply 
tens  by  tens,  and  add  in  what  was  reserved  to  carry, 
thus:  8X5  are  40-f-4  are  44,  now  set  down  the 
whole  amount,  which  is  44. 

EXAMPLE  3.— EXPLANATION. 

5X3  are  15,  set  down  the  5  and  carry  the  43 
1   to  the  next  product;   5X4  are   20=1         25 

are  21;  2x3  are  6,  21+6  are  27,  set  down     

the  7  and  carry  the  2;  2X4  are  8+2  are  1075 
10  ;  now  set  down  the  whole  amount. 

When  the  multiplicand  is  composed  of  three  fig- 
ures, and  there  are  only  two  figures  in  the  multi- 
plier, we  obtain  the  product  by  the  following 

lluLE. — Set  down  the  smaller  factor  under  the 
larger,  units  under  units,  tens  under  tens  ;  now  muU 
tiply  the  first  upper  figure  hy  tlie  unit  figure  of  the 
multiplier,  setting  down  the  units  of  the  product,  and 
reserving  the  tens  to  he  added  to  the  next  product; 
now  multiply  the  second  upper  hy  units,  and  the  first 
upper  hy  tens,  add  these  two  products  together,  set- 
ting down  the  units  figure  of  their  sum,  and  reserv- 
ing the  tens  to  carry,  as  hefore;  now  multiply  the 
third  upper  hy  units,  and  the  sezond  upper  hy  tens, 
add  these  two  products  together,  setting  down  the 
units  figure  of  their  sum,  and  reserving  the  tens  to 


MULTIPLICATION.  23 

carry ^  as  usual ;  now  multiply  the  tJiird  upper  hy 
tens,  add  in  the  reserved  figure,  if  there  is  one,  and 
set  down  tlie  whole  amount.  This  will  he  the  com- 
plete  product. 

Remark. — One  of  the  principal  errors  with  the 
beginner,  in  this  system  of  multiplication,  is 
neglecting  to  add  in  the  reserved  figure.  The  stu- 
dent must  bear  in  mind  that  the  reserved  figure  is 
added  on  to  the  first  product  obtained  after  the  set- 
ting down  of  a  figure  in  the  complete  product. 

EXAMPLE  1.— EXPLANATION. 

Multiply  first  upper  by  units,  5x3  are  123 
15,  set  down  the  5,  reserve  the  1  to  carry         45 

to  the  next  product;  now  multiply  second     

upper  by  units  and  first  upper  by  tens,  5X2  5535 
are  10-|-1  are  11,4X3  are  12,  add  these 
products  together;  ll-j-12  are  23,  set  down  the  3, 
reserve  the  2  to  carry ;  now  multiply  third  upper 
by  units,  and  second  upper  by  tens,  add  these  two 
products  together,  always  adding  on  the  reserved 
"figure  to  the  first  product;  5x1  are  5-|-2  are  7, 
4X2  are  8,  and  7-}-8  are  15,  set  down  the  5,  re- 
serve the  1 ;  now  muitiply  third  upper  by  tens, 
and  set  down  the  whole  amount;  4X1  are  4-|-l  are 
5,  set  down  the  5.  This  will  give  the  comple^« 
product. 


24         orton's  lightning  calculator. 

Multiply  32  by  45  in  a  single  line. 

Here  we  multiply  5X2  and  set  down  and  32 
carry  as  usual ;  then  to  what  you  carry  add         45 

5X3  and  4X  2,  which  gives  24;  set  down    

4  and  carry  2  to  4x3,  which  gives  14  and     1440 
completes  the  product. 

Multiply  123  by  456  in  a  single  line. 

Here  the  first  and  second  places  are  123 
found  as  before;  for  the  third,  add  6Xl>  456 

5X2,  4x3,  with  the  2  you  had  to  carry,  

making  30  ;  set  down  0  and  carry  3 ;  then  56088 
drop  the  units'  place  and  multiply  the 
hundreds  and  tens  crosswise,  as  you  did  the  tens 
and  units,  and  you  find  the  thousand  figure ;  then, 
dropping  both  units  and  tens,  multiply  the  4X1, 
adding  the  1  you  carried,  and  you  have  5,  which 
completes  the  product.  The  same  principle  may 
be  extended  to  any  number  of  places;  but  let  each 
step  be  made  perfectly  familiar  before  advancing 
to  another.  Begin  with  two  places,  then  take  three, 
then  four,  but  always  practicing  some  time  on  each 
number,  for  any  hesitation  as  you  progress  will, 
confuse  you. 

N.  B.  The  following  mode  of  multiplying  num- 
bers will  only  apply  where  the  sum  of  the  two  last 
or  unit  figures  equal  ten,  and  the  other  figures  in 
both  factors  are  the  same. 


MULTIPLICATION.  25 

CONTRACTIONS  IN  MULTIPLICATION. 

To  multiply  when  the  unit  figures  added  equal 
(10)  and  the  tens  are  alike  as  t2  by  IS,  &c. 

1st.  Multiply  the  units  and  set  down  the  result. 

2d.  Add  1  to  either  number  in  tens  place  and 
multiply  by  the  other,  and  you  have  the  complete 
product. 

EXAMPLE  FIRST — PROCESS. 

Here  because  the  sum  of  the  units  4  and  6  86 
are  ten  and  the  tens  are  alike  ;  we  simply  say       84 

4  times  6  are  24,  and  set  down  both  figures  of 

the  product ;  then  because  4  and  6  make  ten  we  '^^24 
add  1  to  8,  making  9,  and  9  times  8  are  72,  which 
completes  the  product. 

Note. — If  the  product  of  units  do  not  contain  ten  the  place 
of  tens  must  be  filled  with  a  cipher 

The  above  rule  is  useful  in  examples  like  the  fol- 

2.  What  will  93  acres  of  land  cost  at  97  dollars 
lowing : 

per  acre  ?  Ans.  $9021. 

3.  What  will  89  pounds  of  tea  cost  at  81  cents 
per  pound  ?  Ans.  $72.09. 

I7i  the  above  the  product  of  9  by  1  did  not  amount 
to  ten,  therefore  0  is  placed  in  tens  place. 

4.  Multiply  998  by  992.  Ans.  990016. 
In  the  above,  because  2  and  8  are  10,  we  add  1  to 

99,  maJcing  100;  then  100  times  99  are  9900. 


26         orton's  lightning  calculator. 

EXAMPLE  EIGHTEENTH. 
Multiply  79  by  71  in  a  single  line. 
Here  we  multiply  IX^  and  set  down  the       79 
result,  then  we  multiply  the  7  in  the  mul-       71 

tiplicand,  increased   by  1  by  the  7  in  the  

multiplier,  7X8,  which  gives  56  and  com-  5609 
pletes  the  product. 

EXAMPLE   NINETEENTH. 

Multiply  197  by  193  in  a  single  line. 

Here  we  multiply  3x7  and  set  down  the  197 
result,    then    we    multiply  the   19  in  the       193 

multiplicand,  increased  by  1  by  the  19  in 

the  multiplier,  19X20,  which  gives  380  38021 
and  completes  the  product. 

EXAMPLE  TWENTIETH. 

Multiply  996  by  994  in  a  single  line. 

Here  we  multiply  4x6  and  set  down  996 
the    result,    then  we  multiply  the  99    in         994 

the  multiplicand,  increased  by  1  by  the  

99  in  the  multiplier,  99X100,  which  990024 
gives  9900  and  completes  the  product. 

EXAMPLE  TWENTY-FIRST. 

Multiply  1208  by  1202  in  a  single  line. 

Here  we  multiply  2x8  and  set  down  1208 
the  result,  then  we  multiply  the  120  in         1202 

the  multiplicand,  increased  by  1  by  the 

120  in  the  multiplier,  120X121,  which  1452016 
gives  14520  and  completes  the  product. 


MULTIPLICATION.  27 

CURIOUS  AND  USEFUL  CONTRACTIONS. 

To  multiply  any  number,  of  two  figures,  by  11, 
HuLE. —  Write  the  sum  of  the  figures  between  them. 

1.  Multiply  45  by  11.  Ans.  495. 
Here  4  and  5  are  9,  which  write  between  4  &  5 

2.  Multiply  34  by  11.  Ans.  374. 
N.  B.  When  the  sum  of  the  two  figures  is  over 

9,  increase  the  left-hand  figure  by  the  1  to  carry. 

3.  Multiply  87  by  11.  Ans.  957. 

To  square  any  number  of  9s  instantaneously, 
and  without  multiplying. 

Rule. —  Write  down  as  many  9s  less  one  as  there 
are  9s  in  the  given  number,  an  8,  as  many  Os  a& 
9«,  and  a  1. 

4.  What  is  the  square  of  9999?    Ans.  99980001. 
Explanation. — We  have  four  9s  in  the  given 

number,  so  we  write   down   three  9s,  then  an  8, 
then  three  Os,  and  a  1. 

5.  Square  999999.  Ans.  999998000001. 

To  square  any  number  ending  in  5, 

KuLE. —  Omit  the  5  and  mnltijyiy  the  number,  as 

it  will  then  stand  by  the  next  higher  number,  and 

annex  25  to  the  product. 

6.  What  is  the  square  of  75  ?  Ans.  5625. 
Explanation. — We  simply  say,  7  times  8  are 

56,  to  which  we  annex  25. 

7.  What  is  the  square  of  95?  Ans.  9025. 

C 


28  ORTON'S    LIGHTNING   CALCULATOR. 

Mental  Operations  in  Fractions. 

To  square  any  number  containing  J,  as  6^,  9^, 

KuLE. — Multiply  the  whole  number  by  the  next 
higher  whole  number ^  and  annex  \  to  the  product. 

Ex.  1.  What  is  the  square  of  7^?     Ans.  hQ{. 

We  simply  say,  7  times  8  are  56,  to  which  we 
addi. 

2.  What  will  ^  lbs.  beef  cost  at  9^  cts.  a  lb.? 

3.  What  will  121  yds.  tape  cost  at  \2\  cts.  a  yd.  ? 

4.  What  will  5^  lbs.  nails  cost  at  5^  cts.  a  lb.  ? 

5.  What  will  11^  yds.  tape  cost  at  11^  cts.  a  yd.? 

6.  What  will  19^  bu.  bran  cost  at  191  cts.  a  bu.? 
Reason. — We  multiply  the  whole  number  by 

the  uext  higher  whole  number,  because  half  of  any 
number  taken  twice  and  added  to  its  square  is  the 
same  as  to  multiply  the  given  number  by  one  moie 
than  itself.  The  same  principle  will  multiply  any 
two  like  numbers  together,  when  the  sum  of  the 
fractions  is  one,  as  8^  by  8|,  or  11|  by  11|,  etc 
It  is  obvious  that  to  multiply  any  number  by  any 
two  fractions  whose  sum  is  one,  that  the  sum  of  the 
products  must  be  the  original  number^  and  adding  the 
number  to  its  square  is  simply  to  multiply  it  by 
one  more  than  itself;  for  instance,  to  multiply  7J 
by  7|,  we  simply  say,  7  times  8  are  56,  and  then, 
to  complete  the  multiplication,  we  add,  of  course, 
the  product  of  the  fractions  (|  times  \  arc  3^), 
maJcing  ^^-^^  the  answer. 


MULTIPLICATION.  29 

Wliere  tlie  sum  of  tlie  Fractions  is  One. 

To  multiply  any  two  like  numbers  together  when 
the  sum  of  the  fractions  is  one. 

IluLE. — Multiply  the  whole  numher  hy  the  next 
higher  whole  numher;  after  which^  add  the  product 
of  the  fractions. 

N.  B.  In  the  following  examples,  the  product  of 
the  fractions  are  obtained y^rs^  for  convenience. 

PRACTICAL  EXAMPLES  FOR  BUSINESS  MEN. 

Multiply  3|  by  3^  in  a  single  line. 

Here  we  multiply  |X|,  which  gives  y^^,  3J 
and  set  down  the  result;  then  we  multiply       3-| 

the    3    in    the    multiplicand,   increased    by  

unity,  by  the  3  in  the  multiplier,  3x4,  12^^ 
which  gives  12  and  completes  the  product. 

Multiply  7f  by  7f  in  a  single  line. 

Here  we  multiply  |Xf)  which  gives  -^^^  7f 
and  set  down  the  result;  then  we  multiply       7| 

the  7  in  the  multiplicand,  increased  by  unity,  

by  the  7  in  the  multiplier,  7x8,  which  gives  66^5 
66,  and  completes  the  product. 

Multiply  11^  by  11|  in  a  single  line. 

Here  we  multiply  f  X^,  which  gives  f ,  and  11 J 
set  down  the  result;  then  we  multiply  the  11     11 1 

in  th«  multiplicand,  increased  by  unity,  by 

the  11  in  the  multiplier,  11x12.  which  gives  1325 
132,  and  completes  the  product. 


30         orton's  lightning  calculator. 

EXAMPLE  THIRTY-THIRD. 

Multiply  16f  by  16J  in  a  single  line. 

Here  we  multiply  JXf  which  gives  I,  and  IGJ 
Bet  down  the  result,  then  we  multiply  the       16J 

16    in    the     multiplicand,    increased     by    

unity  by  the  16  in  the  multiplier,  16X17,     272f 
which  gives  272  and  completes  the  product. 
EXAMPLE  THIRTY-FOURTH. 

Multiply  29 J  by  29 J  in  a  single  line. 

Here  we  multiply  JXj  which  gives  J,  29 J 
and  set  down  the  result,  then  we  multiply       29J 

the  29  in  the  multiplicand,  increased  by    

unity  by  the   29  in  the  multiplier,  29  x     87 OJ 
30,  which  gives  870  and  completes  the  pro- 
duct. 

EXAMPLE  THIRTY-FIFTH. 

Multiply  999f  by  999§  in  a  single  line. 

Here  we  multiply  fXfj  which  gives  999f 

J|,  and  set  down  the  result,  then  we  999f 

multiply  the  999  in  the  multiplicand, 

increased  by  unity  by  the  999  in  the     999000j| 
multiplier,    999x1000,    which    gives 
999000  and  completes  the  product. 

Note. — The  system  of  multiplication  introduced 
in  the  preceding  examples,  applies  to  all  numbers. 
Where  the  sum  of  the  fractions  is  one,  and  the  whole 
numbers  are  alike,  or  differ  by  one^  the  learner  is 
requested  to  study  well  these  useful  properties  of 
numbers. 


orton's  lightninq  calculator.         hi 

WJiere  the  sum  of  the  Fractions  is  One. 

To  multiply  any  two  numbers  whose  difference 
is  onej  and  the  sum  of  the  fractions  is  one, 

Rule. — Multiply  the  largernumher,  increased  hy 
ONE,  hy  the  smaller  number;  then  square  the  frac- 
tion of  the  larger  number,  and  subtrac*  its  square 
from  ONE. 

rRACTICAL  EXAMPLES  FOR  BUSINESS  MEN. 

1.  What  will  9:^  lbs.  sugar  cost  at  8J  cts.  a  lb.? 
Here  we  multiply  9,  increased  by  1,  by  8,         91 

thus,  8X10  are  80,  and  set  down  the  result;         gs 

then  from  1  we  subtract  the  square  of  },     

thus,  \  squared  is  j\,  and  1  less  ^  is  |-|.        8()fJ 

2.  What  will  82  bu.  coal  cost  at  7J  cts.  a  bu.? 
Here  we  multiply  8,  increased  by  1,  by  S| 

7.  thus,  7  times  9  are  63,  and  set  down  the  7^ 

result ;  then  from  1  we  subtract  the  square       ~~^ 
of  |,  thus,  I  squared  is  *,  and  1,  less  ^,  is  |.  » 

3.  What  will  lly2^  bu.  seed  cost  at  $10]^  a  bu.? 
Here  we  multiply  11,  increased  by  1,  by 

10,  thus,  10   times   12  are   120,  and  set     l^A 
down  the  result;  then  from  1  we  subtract     ^^il 
the  square  of  -^3,  thus,  ^^  squared  is  y|^,  ^J^^TTT 
and  1  less  , I,  is  ill.  -^  ^'^ 

4.  IIow  many  square  inches  in  a  floor  99|  in 
wide  and  98|  in.  long?  Ans.  9800^5. 

c* 


32         orton's  lightning  calculator.- 

METHOD  OF  OPERATION. 
EXAMPLE  FIRST. 

Multiply  6 J  by  6 J  in  a  single  line. 

Here  we  add  6J-|-i)  "which  gives  6J ;  this  6} 
multiplied     by  the    6    in    the    multiplier,      6| 

6x6i,  gives  39,  to  which  we  add  the  pro-  

duet  of  the  fractions,  thus  JXi  gives  y'g,  added 
39Jg  to  39  completes  the  product. 

EXAMPLE  SECOND. 

Multiply  lljbylljina  single  line. 

Here  we  would  add  HJ+J,  which  gives  11  J 
12;  this  multiplied  by  the  11  in  the  multi-     il| 

plier  gives  132,  to  which  we  add  the  product  

of  the  fractions,  thus  f  X  J  gives  -j?g,  which  132-^^^ 
added  to  132  completes  the  product. 

EXAMPLE  THIRD. 

Multiply  12 J  by  12f  in  a  single  line. 

Here  we  add  12J-["i)  which  gives  13 J ;  12J 

this  multiplied  by  the  12  in  the  multiplier,  12 J 

12X13J,  gives  159,  to  which  add  the  pro 

duct  of  the  fractions,  thus  fxj  gives  f,  159| 
which  added  to  159  completes  the  product. 


orton's  lightninci  calculator.  33 

Where  the  Fractions  have  a  Lihe  Denominator. 

To  multiply  any  two  lihe  numbers  together,  each 
of  which  has  a  fraction  with  a  lihe  denominator,  as 
H  ^1  4J,  or  11^  by  llf,  or  10|  by  lOJ,  etc. 

Rule. — Add  to  the  multiplicand  the  fraction  of 
the  multiplier^  and  multiply  this  sum  hy  the  whok 
number;  after  which^  add  the  product  of  the  fractions. 

PRACTICAL  EXAMPLES  FOR  BUSINESS  MEN. 
N.  B.  In  the  following  example,  the  sum  of  the  frac- 
tions is  ONE. 

1.  What  will  9|  lbs.  beef  cost  at  9J  cts.  a  lb.? 

The  sum  of  9|  and  \  is  ten,  so  we  simply  9| 
say,  9  times  10  are  90;  then  we  add  the  ^ 
product  of  the  fractions,  \  times  |  are  j^^.       90y*- 

N.  B.  In  the  following  example,  the  sum  of  the  frao 
tions  is  less  than  one. 

2    What  will  8J  yds.  tape  cost  at  8|  cts.  a  yd.  ? 

The  sum  of  8|  and  |  is  8J,  so  we  simply       ^\ 

B&j,  8  times  8J  are  70;    then  we  add  the  _i 

product  of  the  fractions,  J  times  ^  are  -^  or  J.     70-J 

N.  B.  In  the  following  example,  the  sum  of  the  frac- 
tions is  greater  than  one. 

3.  What  will  4|  yds.  cloth  cost  at  $4|  a  yd.? 

The  sum  of  4f  and  |  is  5^,  so  we  simply  4| 
say,  4  times  5J  are  21 ;  then  we  add  the  » 
product  of  the  fractions,  J  times  |  are  |^J.       21|^J 

N.  B.  Where  the  fractions  have  different  denominatora, 
reduce  them  to  a  commoc  denominator. 


S4  ORTONS    LIGnTNINO    CALCUI  ATOR. 

Rapid  Process  of  Multiplying  Mixed  ITumhers. 

A  valuable  and  useful  rule  for  the  accountant  in 
the  practical  calculations  of  the  counting-room. 

To  multiply  any  two  numbers  together,  each  of 
which  involves  the  fraction  ^,  as  7^  by  9^,  etc., 

Rttle. —  To  the  product  of  the  whole  numbers  add 
half  their  sum  plus  \, 

EXAMPLES  FOR  MENTAL  OPERATIONS. 

1.  What  will  3^doz.  eggs  cost  at  7^  cts.  a  doz.? 
Here  the  sura  of  7  and  3  is  10,  and  half  this     31 

sum  is  5,  so  we  simply  say,  7  times  3  are  21     l\ 

and  5  are  2G,  to  which  we  add  4-.  

^  264 

N.  B.  If  the  sum  be  an  odd  number,  call  it  one  less 

to  make  it  even,  and  in  such  cases  the  fraction  must  be  |. 

2.  What  will  \\^  lbs.  cheese  cost  at  9|^  cts.  a  lb.  ? 

3.  What  will  8^  yds.  tape  cost  at  15^  cts.  a  yd.? 

4.  What  will  7^  lbs.  rice  cost  at  13^  cts.  a  lb.? 

5.  What  will  lOi  bu.  coal  cost  at  121  cts.  a  bu.? 
Reason. — In  explaining  the  above  rule,  we  add 

half  their  sum  because  half  of  either  number  added 
to  half  the  other  would  be  half  their  sum,  and  we 
add  \  because  ^  by  ^  is  \.  The  same  principle 
will  multiply  any  two  numbers  together,  each  of 
which  has  the  same  fraction;  for  instance,  if  the 
fraction  was  -J^,  we  would  add  one-third  their  sum ; 
if  |,  we  would  add  three-fourths  their  sum,  etc.; 
and  then,  to  complete  the  multiplication,  we  would 
add,  of  course,  the  product  of  the  fractions. 


MULTIPLICATION.  35 

GENERAL  RULE 
For  multiplying  any  two  numbers  together,  each 

of  which  involves  the  same  fraction. 

To  the  product  of  the  whole  numbers^  add  the 
product  of  their  sum  hy  either  fraction ;  after  which, 
add  the  product  of  their  fractions. 

EXAMPLES  FOR  MENTAL  OPERATIONS. 

1.  What  will  11}  lbs.  rice  cost  at  9|  cts!  a  lb.? 
Here  the  sum  of  9  and  11  is  20,  and  three-     Jif 

fourths  of  this  sum  is  15,  so  we  simply  say,       9| 
9  times  11  are  99  and  15  are  114,  to  which 

111    s 

we  add  the  product  of  the  fractions  (^).  ^* 

2.  What  will  7f  doz.  eggs  cost  at  8|  cts.  a  doz.  ? 

3.  What  will  6|  bu.  coal  cost  at  6  j  cts.  a  bu.  ? 

4.  What  will  45|  bu.  seed  cost  at  3f  dol.  a  bu.? 

5.  What  will  3|  yds.  cloth  cost  at  5f  dol.  a  yd.  ? 

6.  What  will  17f  ft.  boards  cost  at  13|  cts  a  ft.? 

7.  What  will  18}  lbs.  butter  cost  at  18|  cts.  a  lb.  ? 

N.  B.  If  the  product  of  the  sum  by  either  frac- 
tion is  a  whole  number  with  a  fraction,  it  is  better 
to  reserve  the  fraction  until  we  are  through  with 
the  whole  numbers,  and  then  add  it  to  the  product 
of  the  fractions ;  for  instance,  to  multiply  3}  by  7 J , 
we  find  the  sum  of  7  and  3,  whicli  is  10,  and  one- 
fourth  of  this  sum  is  2|;  setting  the  ^  down  in 
some  waste  spot,  we  simply  say,  7  times  3  are  21 
and  2  are  23 ;  then,  adding  the  J  to  the  product  of 
tho  fractions  (jq),  gives  j^,  making  23^,  Ans. 


36  ORTON's   LIGHTNINa  CALCULATOR. 

Rapid  Process  of  Multiplying  all  Mixed  Numhen, 
N.  B.  Let  the  student  remember  that  this  is  a 
general  and  universal  rule. 

GENELAL  RULE. 

To  multiply  any  two  mixed  numbers  together, 

1st.  Multiply  the  whole  numhers  together. 

2d.    Multiply  the  upper  digit  hy  the  lower  fraction. 

3d.    Multiply  the  lower  digit  by  the  upper  fraction. 

4th.  Multiply  the  fractions  together. 

5th.  Add  these  FOUR  products  together. 

N.  B.  This  rule  is  bo  simple,  so  useful,  and  so  true  that 
every  banker,  broker,  merchant,  and  clerk  should  post  it 
up  for  reference  and  use. 

PRACTICAL  EXAMPLES  FOR  BUSINESS  ME?< 
N.  B.  The  following  method  is  recommended  to  begin- 
ners: 

Example.— Multiply  12|  by  9|.  12| 

1st.   We    multiply   the   whole   numbers.       9| 
2d.    Multiply  12  by  f  and  write  it  down.  jqS 
3d.    Multiply    9  by  |  and  write  it  down.       9 
4th.  Multiply    f  by  J  and  write  it  down.       6 
5th.  Add  these  four  products   together,  ___£^ 
and  we  have  the  complete  result.       123 ,4^ 
N.  B.  When  the  student  has  become  familiar 
with  the  above  process,  it  is  better  to  do  the  inter- 
mediate work  in  the  head,  and,  instead  of  setting 
down  the  partial  products,  add  them  in  the  mind 
as  you  pass  along,  and  thus  proceed  very  rapidly. 


MULTIPLICATION.  37 

Multiply  8i  by  lOf 

Here  we  simply  say  10  times  8  are  80       8J 

and  J  of  8  is  2,  making  82,  and  |  of  10  is  lOJ 

2,  which  makes  84;  th«n  ^  times  -J-  is  T^\y  

making  8^2^^  the  answer.  81;^ 

PRACTICAL  BUSINESS  METHOD 
For  Multiplying  all  3Iixed  Nmnhers. 

Merchants,  grocers,  and  business  men  generally, 
in  multiplying  the  mixed  numbers  that  arise  in 
the  practical  calculations  of  their  business,  only 
care  about  having  the  answer  correct  to  the  near- 
est cent;  that  is,  they  disregard  the  fraction. 
When  it  is  a  half  cent  or  more,  they  call  it  an- 
other cent ;  if  less  than  half  a  cent,  they  drop  it. 
And  the  object  of  the  following  rule  is  to  show  the 
business  man  the  easiest  and  most  rapid  process  of 
finding  the  product  to  the  nearest  unit  of  any  two 
numbers,  one  or  both  of  which  involves  a  fraction. 
GENERAL  RULE. 

To  multiply  any  two  numbers  to  the  nearest  unit, 

1st.  Multiply  the  whole  number  in  the  multiplicand 
by  the  fraction  in  the  multiplier  to  the  nearest  unit. 

2d.  Multiply  the  whole  number  in  the  multiplier  by 
the  fraction  in  the  m^ultiplicand  to  the  nearest  unit 

3d.  Multiply  the  whole  numbers  together  and  add 
the  three  products  in  your  mind  as  you  proceed. 

N.  B.  In  actual  business  the  work  can  generally  be 
done  mentally  for  only  easy  fractions  occur  iti  business. 


38  ORTON*S   LIGHTNINQ   CALCULATOR. 

N  B.  This  rule  is  so  simple  and  so  true,  according  to 
all  business  usage,  that  every  accountant  should  make 
himself  perfectly  familiar  with  its  application.  There 
being  no  such  thing  as  a  fractian  to  add  in,  there  is 
scarcely  any  liability  to  error  or  mistake.  By  no  other 
arithmetical  process  can  the  result  be  obtained  by  so  few 
figures. 

EXAMi'LES  FOR  MENTAL  OPERATION. 
EXAMPLE    FIRST. 

Multiply  11^  by  8|  by  business  method.      11 J 
Here  J  of  11  to  the  nearest  unit  is  3,  and  ^  of      8  J 

8  to  the  nearest  unit  is  3,  making  6,  so  we  sim-     ' 

ply  say,  8  times  11  are  88  and  6  are  94,  Ans.     94 

Reason. — \  of  11  is  nearer  3  than  2,  and  J  of  8  is  nearer 
3  than  2.    Make  the  nearest  whole  number  the  quotient. 

EXAMPLE   SECOND. 

Multiply  7|  by  9|  by  business  method. 

Here  |  of  7  to  the  nearest  unit  is  3,  and  J  7| 

of  9  to  the  nearest  unit  is  7 ;  then  3  plus  7  9| 

is  10,  so  we  simply  say,  9  times  7  are  63  and     

10  are  73,  Ans.  73 

EXAMPLE   THIRD. 

Multiply  23^  by  19^  by  business  method. 

Here  ^  of  23  to  the  nearest  unit  is  6,  and  23J 
^  of  19  to  the  nearest  unit  is  6  ;  then  6  plus     19 j 

6  is  12,  so  we  simply  say,  19  times  23  are  

437  and  12  are  449,  Ans.  "^'^^ 

N.  B.  In  multiplying  the  whole  numbers  together,  al- 
ways use  the  single-line  method,  ? 


MULTIPILCATION.  39 

EXAMPLE    rOURTII. 

Multiply  128|  by  25  by  businesb  method. 
Here  |  of  25  to  the  nearest  unit  is  17,  so     12^  J 
wc  simply  say,  25  times  128  are  3200  and ^^ 

17  are  3217,  the  answer.  3217 

PRACTICAL  EXAMPLES  FOR  BUSINESS  MEN. 

1.  What  is  the  cost  of  17^  lbs.  sugar  at  18|  cts. 
per  lb.? 

Here  I  of  17  to  the  nearest  unit  is  13,       17 J 
and  ^  of  18;   is  9  13  plus  9  is  22,  so  we       18| 

simply  say,  18  times  17  are  306  and  22  arc 

328,  the  answer.  ^-^"^ 

2.  What  is  the  cost  of  11  lbs.  5  oz.  of  butter  at 
33 J  cts.  per  lb.? 

Here  ^  of  11  to  the  nearest  unit  is  4,       11^^ 
and  ^  of  33  to  the  nearest  unit  is  10 ;       33J 

then  4  plus  10  is  14,  so  we  simply  say,  33 

times  11  are  3G3,  and  14  are  377,  Ans.       ^'^'^'^ 

3.  What  is  the  cost  of  17  doz.  and  9  eggs  at 
12^  cts.  per  doz.? 

Here  J  of  17  to  the  nearest  unit  is  9,       17^^ 
and  ^Sj  of  12  is  9 ;  then  nine  plus  9  is  18,       12^ 
so  we  simply  say,  12  times  17  are  204  and  

18  are  222,  the  answer.  ^^^^ 

4.  What  will  be  the  cost  of  15J  yds.  calico  at 
12.^  cts.  per  yd.?  Ans.  $1.97. 

N.  B.  To  multiply  by  aliquot  parts  of  100,  see  page  44., 
D 


40         orton's  ligiitninu  calculator. 
RAPID  PROCESS  OF  MARKING  GOODS. 

▲  VALUABLE  HINT  TO  MERCHANTS  ANE  ALL  RETAIL  DKALKRS 
IN  FOREIGN  AND  DOMESTIC  DRY  GOODS. 

Retail  merchants,  in  buying  goods  by  wbolc- 
sale,  buy  a  great  many  articles  by  the  dozen,  such 
as  boots  and  shoes,  hats  and  caps,  and  notions  of 
various  kinds.  Now,  the  merchant,  in  buying,  for 
instance,  a  dozen  hats,  knows  exactly  what  one  of 
those  hats  will  retail  for  in  the  market  where  he 
deals ;  and,  unless  he  is  a  good  accountant,  it  will 
often  take  him  some  time  to  determine  whether  he 
can  afford  to  purchase  the  dozen  bats  and  make  a 
living  profit  in  selling  them  by  the  single  hat ;  and 
in  buying  his  goods  by  auction,  as  the  merchant 
often  does,  he  has  not  time  to  make  the  calculation 
before  the  goods  are  cried  off.  He  therefore  loses 
the  chance  of  making  good  bargains  by  being 
afraid  to  bid  at  random,  or  if  he  bids,  and  the 
goods  are  cried  off,  he  may  have  made  a  poor  bar- 
gain, by  bidding  thus  at  a  venture.  It  then  be- 
comes a  useful  and  practical  problem  to  determine 
instantly  what  per  cent,  he  would  gain  if  he  re- 
tailed the  hats  at  a  certain  price. 

To  tell  what  an  article  should  retail  for  to 
make  a  profit  of  20  per  cent., 

Rule. — Divide  what  the  articles  cost  per  dozen  hy 
10,  which  is  done  hy  removing  the  decimal  point  one 
place  to  the  left. 


/  MOLTIPLICATION.  41 

For  instance,  if  hats  cost  $17.50  per  dozen,  re- 
move the  decimal  point  one  place  to  the  left,  mak- 
ing §1.75,  what  they  should  be  sold  for  apiece  to 
gain  20  per  cent,  on  the  cost.  If  they  cost  $31.00 
per  dozen,  they  should  be  sold  at  $3.10  apiece,  etc. 
We  take  20  per  cent,  as  the  basis  for  the  following 
reasons,  viz. :  because  we  can  determine  instantly, 
by  simply  removing  the  decimal  point,  without 
changing  a  figure;  and,  if  the  goods  would  not 
bring  at  least  20  per  cent,  profit  in  the  home  mar- 
ket, the  merchant  could  not  afford  to  purchase  and 
would  look  for  goods  at  lower  figures. 

Reason. — The  reason  for  the  above  rule  is  ob- 
vious :  For  if  we  divide  the  cost  of  a  dozen  by  12, 
we  have  the  cost  of  a  single  article ;  then  if  we 
wish  to  make  20  per  cent,  on  the  cost,  (cost  being 
{  or  f,),  we  add  the  20  per  cent.,  which  is  ^,  to 
the  |,  making  f  or  |J ;  then  as  we  multiply  the 
cost,  divided  by  12,  by  the  |^  to  find  at  what  price 
one  must  be  sold  to  gain  20  per  cent.,  it  is  evident 
that  the  12s  will  cancel,  and  leave  the  cost  of  a 
dozen  to  be  divided  by  10,  which  is  done  by  re- 
moving the  decimal  point  one  place  to  the  left. 
.1.  If  I  buy  2  doz.  caps  at  $7.50  per  doz.,  what 
shall  I  retail  them  at  to  make  20%?     Ans.  75  cis. 

2.  "When  a  merchant  retails  a  vest  at  $4.50  and 
makes  20%,  what  did  he  pay  per  doz.?    Ans.  $45. 

3.  At  what  price  should  I  retail  a  pair  cf  boots 
that  cost  $85  per  doz.,  to  make  20%?    Ans.  $8.50. 


42         orton's  lightning  calculator. 

I 
RAPID  PROCESS  OF  MARKING  GOODS  AT  DIFFERENT 

PER  CENTS. 

Now,  as  removing  the  decimal  point  one  place 
to  the  left,  on  the  cost  of  a  dozen  articles,  gives 
the  selling  price  of  a  single  one  with  20  per  cent, 
added  to  the  cost,  and,  as  the  cost  of  any  article 
is  100  per  cent.,  it  is  obvious  that  the  selling  price 
would  be  20  per  cent,  more,  or  120  per  cent.; 
hence,  to  find  50  per  cent,  profit,  which  would 
make  the  selling  price  150  per  cent.,  we  would 
first  find  120  per  cent.,  then  add  30  per  cent.,  by 
increasing  it  one-fourth  itself;  to  make  40  per 
cent.,  add  20  per  cent.,  by  increasing  it  one-sixth 
itself;  for  35  per  cent.,  increase  it  one-eighth  itself, 
etc.  Hence,  to  mark  an  article  at  any  per  cent, 
profit,  we  have  the  following 

GENERAL  RULE 

First  find  20  per  cent,  profit^  hy  removing  the 
decimal  point  one  place  to  the  left  on  the  price  the 
articles  cost  a  dozen;  then,  as  20  per  cent,  profit  is 
120  per  cent.y  add  to  or  subtract  from  this  amount 
the  fractional  part  that  the  required  per  cent,  added 
to  100  is  more  or  less  than  120. 

Merchants,  in  marking  goods,  generally  take  a 
per  cent,  that  is  an  aliqot  part  of  100,  as  25^, 
33^%,  50%,  etc.  The  reason  they  do  this  is  be- 
cause it  makes  it  much  easier  to  add  such  a  per 
cent,  to  the  cost;  for  instance,  a  merchant  could 


MULTIPLICATION.  4Z 

mark  almost  a  dozen  articles  at  50  per  cent,  profit 
in  the  time  it  would  take  him  to  mark  a  single 
one  at  49  per  cent.  For  the  benefit  of  the  student, 
and  for  the  convenience  of  business  men  in  mark- 
ing goods,  we  have  arranged  the  following  table : 

TABLE 

For  Marking  all  Articles  hoiight  hy  the  Dozen. 
N.  B.  Most  of  these  are  used  in  business. 
To  make  20%  remove  the  point  one  place  to  the  left. 


(( 

(( 

80% 

(( 

a 

and  add  ^  : 

itself. 

t( 

(t 

60% 

(( 

a 

a 

a 

a 

t( 

a 

50% 

u 

a 

a 

a 

i( 

« 

« 

44% 

(( 

a 

(( 

a 

u 

(( 

t( 

40% 

(( 

(t 

a 

a 

t( 

u 

i( 

37^% 

(( 

a 

a 

a 

(t 

u 

it 

35% 

(( 

a 

a 

i( 

(i 

il 

« 

33  J  % 

(( 

a 

a 

a 

a 

« 

« 

32% 

(( 

a 

a 

a 

A 

i( 

u 

(( 

30% 

a 

a 

i( 

a 

A 

i( 

n 

<( 

28% 

(( 

a 

(( 

a 

A 

<( 

(< 

(( 

26% 

u 

a 

a 

a 

A 

(( 

u 

(( 

25% 

« 

a 

a 

(( 

A 

it 

(t 

(i 

121% 

a 

a 

subtract  yV 

a 

11 

(( 

10|% 

a 

(( 

a 

A 

a 

tt 

(( 

18|% 

u 

a 

a 

a 

A 

a 

If  I  buy  1  doz.  shirts  for  $28.00,  what  shall  I 
retail  them  for  to  make  50%?  Ans.  $3.50. 

Explanation. — Remove  the  point  one  place  to 
the  left,  and  add  on  \  itself. 


44 


ORTON  S    LIGHTNING    CALCULATOR. 


Where  the  Multiplier  is  an  Aliquot  Part  of  100. 

Merchants  in  selling  goods  generally  make  the 
price  of  an  article  some  aliquot  part  of  100,  as  in 
selling  sugar  at  12J  cents  a  pound  or  8  pounds 
for  1  dollar,  or  in  selling  calico  for  16|  cents  a 
yard  or  6  yards  for  1  dollar,  etc.  And  to  be- 
come familiar  with  all  the  aliquot  parts  of  100,  so 
that  you  can  apply  them  readily  when  occasion 
requires,  is  perhaps  the  most  useful,  and,  at  the 
same  time,  one  of  the  easiest  arrived  at  of  all  the 
computations  the  accountant  must  perform  in  the 
practical  calculations  of  the  counting-room. 

TABLE  OF  THE  ALIQUOT  PARTS  OF  100  AND  1000 
N.  B.  Most  of  these  are  used  in  business. 


l^i 

is  \  part  of  100. 

H 

is  ^^  part  of    100. 

25 

8  1  or  ^  of  100. 

IGf 

is  ^^ovloi    100. 

37i 

s  1  part  of  100. 

33| 

is  i*^or|of    100. 

50 

18  1  or  ^  of  100. 

66| 

is  ^j  or  1  of    100. 

62^ 

8  1  part  of  100. 

83  i 

isfgorfof    100. 

75    ] 

8  f  or  f  of  100. 

125 

is  I  part  of  1000. 

87| 

is   I  part  of  100. 

250 

is  1  or  ^  of  1000. 

Gi^ 

8  tV  part  of  1<^<^- 

375 

is  1  part  of  1000. 

18| 

s  -^^  part  of  100. 

625 

is  f  part  of  1000. 

'6\\. 

8  T^g.  part  of  100. 

875 

is   I  part  of  1000. 

To  multiply  by  an  aliquot  part  of  100, 

Rule — Add  two  ciphers  to  the  multiplicand,  thp.n 

take  such  part  of  it  as  the  rmdtiplier  is  part  of  100. 
N.  B.  If  the  multiplicand  is  a  mixed  number  reduce 

the  fraction  to  a  decimal  of  two  places  before  dividing. 


COUNTIXG-ROOM  EXERCISES. 


Examples. — 1.  Multiply  424  by  25. 

As  25  =  J  of  100,  divide  42400  by  4  =  10600. 

N.  B.  If  the  multiplicand  is  a  mixed   number,  reduce  the 
fraction  to  a  decimal  of  two  places  before  dividing. 

2.  Give  the  cost  of  12|-  yds.  cloth  @  18|c.  per  yd. 
Process. — 12J  =  ^;  changing  18J  to  a  decimal, 

we  have  18.75  ^  8  =  $2.34f. 

Note. — Aliquot  pnrts  may  be  conveniently  used  when  the  mul- 
tiplier is  but  little  more  or  less  than  an  aliquot  part. 

3.  Multiply  24  by  ITf. 

1st.  Multiply  24  by  16f  (the  one-sixth  of  100). 
Thus    24    X    16f    =  2400  ^  6  =  400 
As  17|  =  16|  -f  1  multiply  24  by  1  =  24 
Kcncc  24  X  17§  =  the  two  products,  424 
45 


46         orton's  liohtning  calculator. 

Rationale. — As  in  the  last  case,  by  annexing 
two  ciphers,  we  increase  the  multiplicand  one  hun- 
dred times ;  and  by  dividing  the  number  by  3,  we 
only  increase  the  multiplicand  thirty-three  and 
one-third  times,  because  33 J  is  one-third  of  100. 

4.  To  multiply  any  number  by  333J  add  three 
ciphers,  and  divide  by  3. 

Multiply  4797  by  333J.      Product,  1599000. 

3)4797000 

1599000 

5.  To  multiply  any  numbpr  by  6§  add  two  ci- 
phers, and  divide  by  15 ;  or  add  one  cipher  and 
multiply  by  §. 

Multiply  156G  by  6f . 

15)156600 

10440  First  method. 

15660 
2 

3)31320 

10440  Second  method. 

6.  To  multiply  any  number  by  66f  add  three 
ciphers,  and  divide  by  15  j  or  add  two  ciphers  and 
multiply  by  §. 


MULTirLICATION.  47 


Multiply  36G3  by  CG|. 

15)3663000 

244200  First  method. 

366300 
2 

3)732600 

244200  Second  method. 
7.     To  multiply  any  number  by  8J  add  two  ci- 
phers, and  divide  by  12. 

Multiply  2889  by  8J.     Product,  24075. 
12^288900 


24075 
8.     To  multiply  any  number  by  83J   add  three 
ciphers,  and  divide  by  12. 

Multiply  7695  by  83J.     Troduct,  641250. 
12)7095000 


641250 
9.     To  multiply  any  number  by  6J  add  two  ci- 
phers, and  divide  by  16  or  its  factors — 4X4. 
Multiply  7696  by  6J.     Product,  48100. 
4)769600 


4)192400 
48100 


MEASUREMENT  OF  LUMBER. 


The  unit  of  board  measure  is  a  square  foot  1 
inch  thick. 

To  measure  inch  boards. 

Rule. — Multiply  the  length  of  the  board  in  feet 
by  its  breadth  in  inches,  and  divide  the  product 
by  12 ;  the  quotient  is  the  contents  in  square  feet. 

Note. — ^When  the  board  is  wider  at  one  end  than 
the  other,  add  the  width  of  the  two  ends  together, 
and  take  half  the  sum  for  a  mean  width. 

Example. — How  many  square  feet  in  a  board 
10  feet  long,  13  inches  wide  at  one  end,  and  9 
inches  wide  at  the  other  ? 

Process. — (13  +  9)  -I-  2  =  11  (mean  width)  then 
10  length  X  11  =  110  -f.  12  =  9^  feet.     Ans. 
48 


MEASUREMENT  OP  LUMBER.  49 

Sawed  lumber,  as  joists,  plank,  and  scantlings, 
are  now  generally  bought  and  sold  by  board  mea- 
sure. The  dimensions  of  a  foot  of  board  measure 
are  1  foot  long,  1  foot  wide,  and  1  inch  thick. 

To  ascertain  the  contents  (board  measure)  of 
boards,  scantling,  and  plank. 

Rule. — Multiply  the  width  in  inches  by  the 
thickness  in  inches,  and  that  product  by  the  length 
in  feet,  which  last  product  divide  by  12. 

Example. — How  many  feet  of  lumber  in  14 
planks  16  feet  long,  18  inches  wide,  and  4  inches 
thick  ? 

Process. — 16  feet  X  18  inches  X  4  inches  =  1152, 
then  1152 -T- 12  =  96  feet  ==  (one  plank)  X  14  = 
134  feet.     Ans. 

2.  To  ascertain  the  quantity  of  lumber  in  a  log, 
or  its  board  measure. 

Rule.* — Multiply  the  diameter  in  inches  at  the 
small  end  by  one-half  the  number  of  inches,  and 
this  product  by  the  length  of  the  log  in  feet,  which 
last  product  divide  by  12. 

Example. — How  many  feet  of  lumber  can  be 
made  from  a  log  which  is  36  inches  in  diameter, 
and  10  feet  long? 

Solution.— 36  X  18  =  648  ;  648  X  10  =  6480  ; 
6480 -T- 12  =  540.     Ans. 

*  The  above  rule  is  the  one  used  in  the  great  pineries  of 
Northern  Michigan,  Wisconsin,  and  Minnesota. 


50 


ORTON  S  LIGHTNING  CALCULATOR. 


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A  Log  Table. — Showing  the  number  of  feet  of 
boards  any  log  will  make  whose  diameter  is  from 
15  to  36  inches  at  the  smallest  end,  and  from  10 


to  15  feet  in  length. 


MEASUREMENT  OF  WOOD. 


Wood  is  measured  by  the  cord,  which  contains 
128  cubic  feet. 

Wood  is  bought  and  sold  by  the  cord  and  frac- 
tions of  a  cord. 

Pine  and  spruce  spars  from  10  to  4  inches  in 
diameter  inclusive,  are  measured  by  taking  the  dia- 
meter, clear  of  bark,  at  one-third  of  their  length 
from  the  large  end. 

Spars  are  usually  purchased  by  the  inch  diame- 
ter ;  all  under  4  inches  are  considered  poles. 

Spruce  spars  of  T  inches  and  less,  should  have 
5  feet  in  length  for  every  inch  in  diameter. 
E  61 


52         orton's  lightning  calculator. 

Note. — A  pile  of  wood  that  is  8  feet  long,  4  feet 
higL,  and  4  feet  wide,  contains  128  cubic  feet,  or 
a  cord,  and  every  cord  contains  8  cord-feet;  and 
as  8  is  y'g  of  128,  every  cord-foot  contains  16  cubio 
feet ;  therefore,  dividing  the  cubic  feet  in  a  pile  of 
wood  by  16,  the  quotient  is  the  cord-feet ;  and  if 
cord-feet  be  divided  by  8,  the  quotient  is  cords. 

Note. — If  we  wish  to  find  the  circumference  of 
a  tree,  which  will  hew  any  given  number  of  inches 
square,  we  divide  the  given  side  of  the  square  by 
.225,  and  the  quotient  is  the  circumference  re- 
quired. 

What  must  be  the  circumference  of  a  tree  that 
will  make  a  beam  10  inches  square  ? 

Note. — When  wood  is  "  corded"  in  a  pile  4  feet 
wide,  by  multiplying  its  length  by  its  hight,  and 
dividing  the  product  by  4,  the  quotient  is  the  cord- 
feet  ;  and  if  a  load  of  wood  be  8  feet  long,  and  its 
hight  be  multiplied  by  its  width,  and  the  product 
divided  by  2,  the  quotient  is  the  cord-feet. 

How  many  cords  of  wood  in  a  pile  4  feet  wide, 
70  feet  6  inches  long,  and  5  feet  3  inches  high  ? 

Note. — Small  fractions  rejected. 

To  find  how  large  a  cube  may  be  cut  from  any 
given  sphere,  or  be  inscribed  in  it. 

B-ULE. — Square  the  diameter  of  the  sphere,  divide 
that  product  hy  3,  and  extract  the  square  root  of  the 
quotient  for  the  answer^ 


MENSURATION  OR  PRACTICAL  GEOMETR?".        53 

I  have  a  piece  of  timber,  30  inches  in  diameter ; 
how  large  a  square  stick  can  be  hewn  from  it? 

Rule. — Multiply  the  diameter  hy  .7071,  and  the 
product  is  the  side  of  a  square  inscribed, 

I  have  a  circular  field,  360  rods  in  circumference; 
what  must  be  the  side  of  a  square  field  that  shali 
contain  the  same  quantity? 

Rule. — Multiply  the  circumference  hy  .282,  and 
the  product  is  the  side  of  an  equal  square. 

I  have  a  round  field,  50  rods  in  diameter;  what 
is  the  side  of  a  square  field  that  shall  contain  the 
same  area?  Ans.  44.31 135-{- rods. 

Rule. — Multiply  the  diameter  hy  .886,  and  th« 
product  is  the  side  of  an  equal  square. 

There  is  a  certain  piece  of  round  timber,  30 
inches  in  diameter ;  required  the  side  of  an  equi- 
lateral triangular  beam  that  may  be  hewn  from  it. 

Rule. — Multij^ly  the  diameter  hy  .866,  and  the 
product  is  the  side  of  an  inscribed  equilateral  tri- 
angle. 

To  find  the  area  of  a  globe  or  sphere. 

Definition. — A  sphere  or  globe  is  a  round  solid 
body,  in  the  middle  or  center  of  which  is  an  imag- 
inary point,  from  which  every  part  of  the  surface 
is  equally  distant.  An  apple,  or  a  ball  used  by 
children  in  some  of  their  pastimes^  may  be  called 
a  sphere  or  globe. 


ROUND  TBIBER. 


Round  timber,  when  squared,  is  estimated  to 
lose  one-fifth  ;  hence  (50  cubic  feet,  or)  a  ton  of 
round  timber  is  said  to  contain  only  40  cubic  feet. 

Round,  sawed,  and  hewn  timber  is  bought  and 
sold  by  the  cubic  foot. 

To  measure  round  timber. 

Rule.* — Take  the  girth  in  feet,  at  both  the  large 
and  small  ends,  add  them,  and  divide  their  sum  by 
two  for  the  mean  girth ;  then  multiply  the  length 
in  feet  by  the  square  of  one-fourth  of  the  mean 
girth,  and  the  quotient  will  be  the  contents  in  cubic 
feet,  according  to  the  common  practice. 

*  This  rule  gives  ti\io\ii  fmir-ffths  of  the  true  contents,  one- 
fifth  being  allowed  to  the  buyer  for  waste  in  hewing. 

54 


TIMBER  MEASURE.  55 

Example. — What  are  the  cubic  contents  of  a 
round  log  20  feet  long,  9  feet  girth  at  the  large 
end,  and  t  feet  at  the  small  end  ? 

Solution. — 9  +  7  =  16-7-2  =  8  mean  girth. 

Then  20  length  x  4  feet  (the  square  of  ^  mean 
girth)  =  80  cubic  feet.     Aiis. 

Note. — If  the  girth  be  taken  in  inches,  and  the 
length  in  feet,  divide  the  last  product  by  144. 

Example. — What  are  the  cubic  contents  of  a 
round  log  12  feet  long,  50  inches  girth  at  the  large 
end,  38  inches  at  the  small  end? 

Work.— 50  +  38  =  88  -^  2  =  44  mean  girth. 

Then  12  length  X  121  inches  (the  square  of  i 
mean  girth)  =  1452  -^  144  =  lOjV  cubic  feet. 

To  measure  round  timber  as  the  frustum  of  a 
cone:  that  is,  to  measure  all  the  timber  in  the  log. 

Rule. — Multiply  the  square  of  the  circumference 
at  the  middle  of  the  log  in  feet  by  8  times  the  length, 
and  the  product  divided  by  100  will  be  the  contents. 
Extremely  near  the  truth. 

Note. — The  above  rule  makes  1  foot  more  timber 
in  every  190  cubic  feet  a  log  contains  if  ciphered 
out  by  the  long  and  tedious  rules  of  Geometry.  It 
is  therefore  suflScicntly  correct  for  all  practical  pur- 
poses, and  this  rule  being  so  short  and  simple  in 
comparison  with  all  others,  every  lumberman,  ship- 
builder, carpenter,  inspector  or  surveyor  of  timber, 
should  post  it  up  for  reference  and  use. 

E* 


56 


ORTON  S    LIOHTNINQ    CALCULATOR. 


A  TABLE  FOR  MEASURING  TIMBER. 

Quarter 
Girt. 

Area. 

Quarter 
Girt. 

Area. 

Quarter 
Girt. 

Area. 

Incbea. 

Feet. 

Inches. 

Feet. 

luches. 

Feet. 

6 

.250 

12 

1.000 

18 

2.250 

6} 

.272 

12} 

1.042 

18^ 

2.376 

6^ 

.294 

12^ 

1.085 

19 

2.506 

6f 

.317 

121 

1.129 

19^ 

2.640 

7 

.340 

13 

1.174 

20 

2.777 

n 

.364 

13} 

1.219 

20^ 

2.917 

7J 

.390 

13J 

1.265 

21 

3.062 

7| 

.417 

13f 

1.313 

21J 

3.209 

8 

.444 

14 

1.361 

22 

3.362 

8} 

.472 

14} 

1.410 

22^ 

3.516 

8i 

.501 

14* 

1.460 

23 

3.673 

81 

.531 

141 

1.511 

23J 

3.835 

9 

.562 

15 

1.562 

24 

4.000 

n 

.594 

15} 

1.615 

24^ 

4.168 

n 

.626 

15J 

1.668 

25 

4.340 

9| 

.659 

151 

1.722 

25J 

4.516 

10 

.694 

16 

1.777 

26 

4.694 

10} 

.730 

16} 

1.833 

26J 

4.876 

m 

.766 

16* 

1.890 

27 

5.062 

101 

.803 

161 

1.948 

27J 

5.252 

11 

.840 

17 

2.006 

28 

5.444 

lU 

.878 

17} 

2.066 

28^ 

5.640 

m 

.918 

17^ 

2.126 

29 

5.840 

111 

.959 

17f 

2.187 

29J 
30 

6.044 
6.250 

To  measure  round  timber  by  the  table. 
Multiply  the  area  corresponding  to  the  quarter- 
girt  in  inches  by  the  length  of  the  log  in  feet. 


TIMBER  MEASURE. 


57 


Note. — If  the  quarter-girt  exceed  the  table,  take 
half  of  it,  and  four  times  the  contents  thus  formed 
will  be  the  answer. 

EXAMPLE  1. 

If  a  piece  of  round  timber  be  18  feet  long,  and 
the  quarter  girt  24  inches,  how  many  feet  of  timber 
are  contained  therein? 


24  quarter  girt. 
24 

96 

48 

By  the  Tahle. 

576  square. 
18 

Against  24  stands    4  00 
Length,        18 

4608 
576 

Product,                  72.00 
Anc?   72  fppt 

144)10368(72  feet 
1008 

XJlIIO*       1    ^      Iv/VfUs 

288 
288 

This  table  gives  the  customary,  but  only  a.hout  four  Jifths  of 
the  true  contents,  one-fifth  being  allowed  the  buyer  for  waste 
in  hewing  or  sawing  to  make  the  timber  square. 

The  following  rule  gives  the  true  contents  : — 
Multiply  square  of  girth  by  .08  times  length. 
In  the  above  Example  the  whole  girth  is  8  feet, 
s.iuared  is  64  x  (.08  X  18  length)  =  92.16  feet. 


68         orton's  lightning  calculator. 

1.   Of  Flooring. 

Joists  are  measured  by  multiplying  tlieir  breadtli 
b^  their  depth,  and  that  product  by  their  length. 
They  receive  various  names,  according  to  the  posi- 
tion in  which  they  are  laid  to  form  a  floor,  such  as 
trimming  joists,  common  joists,  girders,  binding 
joists,  bridging  joists  and  ceiling  joists. 

Girders  and  joists  of  floors,  designed  to  bear 
great  weights,  should  be  let  into  the  walls  at  each 
end  about  two-thirds  of  the  wall's  thickness. 

In  boarded  flooring,  the  dimensions  must  be 
taken  to  the  extreme  parts,  and  the  number  of 
squares  of  100  feet  must  be  calculated  from  these 
dimensions.  Deductions  must  be  made  for  stair- 
cases, chimneys,  etc. 

Example  1.  If  a  floor  be  57  feet  3  inches  long, 
and  28  feet  6  inches  broad,  how  many  squares  of 
flooring  are  there  in  that  room  ? 
By  Decimals. 
57.25 
28.5 


28G25 
45800 
11450 


By  Duodccimah 

F. 

I. 

57 

:    3 

28 

:    6 

456 

114 

28 

:    7    : 

6 

7 

;    0    : 

0 

100)1631.625  feet. 

Squares  16.31625  16:31   :    7    :    6 

Am.  IQ  squares  and  31  feet. 


SQUARE  TIMBER. 


To  measure  square  timber. 

Rule. — Multiply  the  breadth  in  feet  by  the 
depth  in  feet,  and  that  by  the  length  in  feet,  and 
the  quotient  will  be  the  contents  in  cubic  feet. 

Example. — How  many  cubic  feet  in  a  square  log 
12  feet  long  by  2  feet  broad  and  IJ  feet  deep  ? 

Explanation. — 2  feet  breadth  x  1^  feet  depth 
X  12  feet  length  =  36  cubic  feet.     Ans. 

Note. — If  the  breadth  and  depth  be  taken  in 
inches,  divide  the  last  product  by  144. 

Example. — How  many  cubic  feet  in  a  square  log 
24  feet  long,  30  inches  broad,  and  20  inches  deep  ? 

Solution. — 30  inches  breadth  x  20  inches  depth 
X  24  feet  length  =  14400  -k- 144  =  100  cubic  feet. 
59 


60         obion's  lightninq  calculatoe. 

PROBLEM  III. 

To  find  the  solid  contents  of  squared  or  four-sided 

Timber. 

By  the  Carpenters^  Rule, 

As  12  on  D  :  length  on  c  :  Quarter  girt  on  D  : 

solidity  on  c. 

Rule  I. — Multiply  the  breadth  in  the  middle  by 
the  depth  in  the  middle^  and  that  product  by  the 
length  for  the  solidity. 

Note. — If  the  tree  taper  regularly  from  one  end 
to  the  other,  half  the  sum  of  the  breadths  of  tho 
two  ends  will  be  the  breadth  in  the  middle,  and 
half  the  sum  of  the  depths  of  the  two  ends  will  bo 
the  depth  in  the  middle. 

Rule  II. — Multiply  the  sum  of  the  breadths  of 
the  two  ends  by  the  sum  of  (he  depths,  to  which  add 
the  product  of  the  breadth  and  depth  of  each  end; 
one-sixth  of  this  sum  multiplied  by  the  length,  will 
give  the  correct  solidity  of  any  piece  of  squared  tim- 
ber tapering  regularly. 

PROBLEM  IV. 
To  find  how  much  in  length  will  make  a  solid  foot, 

or  any  other  asssigned  quantity,  of  squared  timber^ 

of  eqtial  dimensions  from  end  to  end. 

Rule. — Divide  1728,  the  solid  inches  in  a  foot, 
or  the  solidity  to  be  cut  off,  by  the  area  of  the  end 
in  inches,  and  the  quotient  will  be  the  length  in  inches. 


TIMBER   MEASURE.  61 

Note. — To  answer  the  purpose  of  the  above 
rule,  some  carpenters'  rules  have  a  little  table  upoii 
them,  in  the  following  form,  called  a  iahle  of  tun- 
her  measure. 


r^ 

0 

0 

|0|9|0 

11 

|3 

9 

inches. 

144 

36 

16 

|9|5|4 

2 

|2 

1 

feet. 

1 1 

2 

3 

|4|5|6 

7 

|8 

9 

side  of  the  square. 

This  table  shows,  that  if  the  side  of  the  square 
be  1  inch,  the  length  must  be  144  feet ;  if  2  inches 
be  the  side  of  the  square,  the  length  must  be  36 
feet,  to  make  a  solid  foot. 


MEASUKEMENT  OF  HAY. 


The  only  correct  mode  of  measuring  hay  is  to 
weigh  it.  This,  on  account  of  its  bulk  and  cha- 
racter, is  very  diflScult,  unless  it  is  baled  or  other- 
wise compacted.  This  difficulty  has  led  farmers 
to  estimate  the  weight  by  the  bulk  or  cubic  con- 
tents, a  mode  which  is  only  approximately  correct. 
Some  kinds  of  hay  are  light,  while  others  are 
heavy,  their  equal  bulks  varying  in  weight.  But 
for  all  ordinary  farming  purposes  of  estimating  the 
amount  of  hay  in  meadows,  mows,  and  stacks,  the 
following  rules  will  be  found  sufficient : — 

As  nearly  as  can  be  ascertained,  25  cubic  yards 
of  average  meadow  hay,  in  windrows,  make  a  ton. 
62 


MEASUREMENT  OF  HAY.  63 

When  loaded  on  wagons,  or  stored  in  barns,  20 
cubic  yards  make  a  ton. 

When  well  settled  in  mows,  or  stacks,  15  cubic 
yards  make  a  ton. 

Note. — These  estimates  are  for  medium-sized 
mows  or  stacks ;  if  the  hay  is  piled  to  a  great 
height,  as  it  often  is  where  horse  hay-forks  are 
used,  the  row  will  be  much  heavier  per  cubic  yard. 

When  hay  is  baled,  or  closely  packed  for  ship- 
ping, 10  cubic  yards  will  weigh  a  ton. 

To  find  the  number  of  tons  in  long  square  stacks. 

Rule.— Multiply  the  length  in  yards  by  the 
width  in  yards,  and  that  by  half  the  altitude  in 
yards,  and  divide  the  product  by  15. 

Example. — How  many  tons  of  hay  in  a  square 
stack  10  yards  long,  5  wide,  and  9  high  ? 

Solution.— 10  X  5  X  4J  =  225  -?-  15  ==  15  tons. 
Ans. 

To  find  the  number  of  tons  in  circular  stacks. 

Kule. — Multiply  the  square  of  the  circumference 
in  yards  by  4  times  the  altitude  in  yards,  and  di- 
vide by  100 ;  the  quotient  will  be  the  number  of 
cubic  yards  in  the  stack  j  then  divide  by  15  for  the 
number  of  tons. 

Example. — How  many  tons  of  hay  in  a  circular 
stack,  whose  circumference  at  the  base  is  25  yards, 
and  height  9  yards  ? 
F 


64  orton's  lightning  calculator. 

Solution. — 25  x  25  =  625,  the  square  of  the 
circumference ;  then  625  x  36  (four  times  the 
length),  =  225000  ^  100  =  225  (the  number  of 
cubic  yards),  then  225  H- 15  =  15,  the  number  of 
tons. 

An  easy  mode  of  ascertaining  the  value  of  a 
given  number  of  lbs.  of  hay,  at  a  given  price  per 
ton  of  2000  lbs. 

Rule. — Multiply  the  number  of  pounds  of  hay 
(coal,  or  anything  else  which  is  bought  and  sold 
by,  the  ton)  by  one-half  the  price  per  ton,  pointing 
off  three  figures  from  the  right  hand  ;  the  remain- 
ing figures  will  be  the  price  of  the  hay  (or  any 
article  by  the  ton). 

Example. — What  will  658  lbs.  of  hay  cost,  @ 
$1  60  per  ton  ? 

Solution. — $T  50  divided  by  2  equals  $3  75,  by 
which  multiply  the  number  of  pounds,  thus  :  658  x 
$3  16  =  246.  Y50,  or  $2  46.     Ans. 

Kote. — The  principle  in  this  rule  is  the  same  as 
in  interest — dividing  the  price  by  two  gives  us  the 
price  of  half  a  ton,  or  1000  lbs.  ;  and  pointing  off 
three  figures  to  the  right  is  dividing  by  1000. 

A  truss  of  hay,  new,  is  60  lbs. ;  old,  56  lbs. ; 
straw,  40  lbs. 

A  load  of  hay  is  36  trusses. 

A  bale  of  hay  is  300  lbs. 


RULES  FOR  DETERMINING  THE  WEIGHT 
OF  LIYE  CATTLE. 


For  cattle  of  a  girth  of  from  5  to  T  feet,  allow 
23  lbs.  to  the  superficial  foot. 

For  cattle  of  a  girth  of  from  t  to  9  feet,  allow 
31  lbs.  to  the  superficial  foot. 

For  small  cattle  and  calves  of  a  girth  of  from  3 
to  5  feet,  allow  16  lbs.  to  the  superficial  foot. 

For  pigs,  sheep,  and  all  cattle  measuring  less  than 
3  feet  girth,  allow  11  lbs.  to  the  superficial  foot. 

Measure  in  inches  the  girth  round  the  breast, 
just  behind  the  shoulder-blade,  and  the  length  of 
the  back  from  the  tail  to  the  forepart  of  the  shoul- 
der-blade. Multiply  the  girth  by  the  length,  and 
divide  hj  144  for  the  superficial  feet,  and  then  mul- 
65 


66  orton's  lightning  calculator. 

tiply  the  superficial  feet  by  the  number  of  lbs. 
allowed  for  cattle  of  different  girths,  and  the  pro- 
duct will  be  the  number  of  lbs.  of  beef,  veal,  or  pork, 
in  the  four  quarters  of  the  animaL  To  find  the 
number  of  stone,  divide  the  number  of  lbs.  by  14. 

Example. — What  is  the  estimated  weight  of 
beef  in  a  steer,  whose  girth  is  6  feet  4  inches,  and 
length  5  feet  3  inches  ? 

Solution. — *IQ  inches  girth,  x  63  inches  length, 
=  4188  -4-  144  =  33J  square  feet,  X  23  =  Y64| 
lbs.,  or  54f  stone.     Ans. 

Note. — When  the  animal  is  but  half  fattened,  a 
deduction  of  one  lb.  in  every  20  must  be  made ; 
and  if  very  fat,  one  lb.  for  every  20  must  be  a^ded. 

Where  great  numbers  of  cattle  are  anaually 
bought  and  sold  under  circumstances  that  forbid 
ascertaining  their  weight  with  positive  accuracy, 
the  estimated  weight  may  be  thus  taken  with  ap- 
proximate exactness — at  least  with  as  much  accu- 
racy as  is  necessary  in  the  aggregate  valuation  of 
stock.  No  rules  or  tables  can,  however,  be  at  all 
times  implicitly  relied  on,  as  there  are  many  cir- 
cumstances connected  with  the  build  of  the  animal, 
the  mode  of  fattening,  its  condition,  breed,  &c., 
that  will  influence  the  measurement,  and  conse- 
quently the  weight.  A  person  skilled  in  estimat- 
ing the  weight  of  stock  soon  learns,  however,  to 
make  allowance  for  all  these  circumstances. 


TO  MEASURE  CORN  ON  THE  COB  IN  CRIBS. 


Corn  is  generally  put  up  in  cribs  made  of  rails ; 
but  the  rule  will  apply  to  a  crib  of  any  size  or  kind, 
whether  equilateral,  or  flared  at  the  sides. 

IVhen  the  crib  is  equilateral. 

Rule.  —  Multiply  the  length  in  feet  by  the 
breadth  in  feet,  and  that  again  by  the  height  in 
feet,  which  last  product  multiply  by  .63  (the  frac- 
tional part  of  a  heaped  bushel  in  a  cubic  foot),  and 
the  result  will  be  the  heaped  bushels  of  ears.  For 
the  number  of  bushels  of  shelled  corn  multiply  by 
.42  (two-thirds  of  .63),  instead  of  .63. 
i*  67 


68  orton's  lightning  calculator. 

Example. — Required  the  number  of  bushels  of 
shelled  corn  contained  in  a  crib  of  ears,  15  feet 
long,  by  5  feet  wide,  and  10  feet  high  ? 

15  length  X  5  width,  x  10  height  =  T50  cubic 
feet.  Then  "750  X  .63  =  4^2.50  heaped  bushels  of 
ears.  Also  T50  X  .42  =  315  bushels  of  shelled 
corn. 

In  measuring  the  height,  of  course,  the  height  of 
the  corn  is  intended.  And  there  will  be  found  to 
be  a  difference  in  measuring  corn  in  this  mode,  be- 
tween fall  and  spring,  because  it  shrinks  very  much 
in  the  winter  and  spring,  and  settles  down. 

Wlien  the  crib  is  flared  at  the  sides. 

Rule. — Multiply  half  the  sum  of  the  top  and  bot- 
tom widths  in  feet  by  the  perpendicular  height  in 
feet,  and  that  again  by  the  length  in  feet,  which 
last  product  multiply  by  .63  for  heaped  bushels  of 
ears,  and  by  .42  for  the  number  of  bushels  of 
shelled  corn. 

Note. — The  above  rule  assumes  that  three  heap- 
ing half  bushels  of  ears  make  one  struck  bushel 
of  shelled  corn.  This  proportion  has  been  adopted 
upon  the  authority  of  the  major  part,  of  our  best 
agricultural  journals.  Nevertheless,  some  journals 
claim  that  two  heaping  bushels  of  ears  to  one  of 
shelled  corn  is  a  more  correct  proportion,  and  it  is 
the  custom  in  many  parts  of  the  country  to  buy 


MULTIPLICATION   AND   DIVISION.  69 

and  sell  at  that  rate.  Of  course  much  will  de- 
pend upon  the  kind  of  corn,  the  shape  of  the  ear, 
the  size  of  the  cob,  &c.  Some  samples  are  to  be 
found,  three  heaping  half  bushels  of  which  will 
even  overrun  one  bushel  shelled ;  while  others 
again  are  to  be  found,  two  bushels  of  which  will  fall 
short  of  one  bushel  shelled.  Every  farmer  must 
judge  for  himself,  from  the  sample  on  hand,  whether 
to  allow  one  and  a  half  or  two  bushels  of  ears  to 
one  of  shelled  corn.  In  either  case,  it  is  only  an 
approximate  measurement,  but  sufficient  for  all  ordi- 
nary purposes  of  estimation.  The  only  true  way 
of  measuring  all  such  products  is  by  weight. 

Multiplication  and  Division, 
To  multiply  one-half,  is  to  take  the  multiplicand 
one-half  of  one  time;  that  is,  take  one-half  of  it, 
or  divide  it  by  2. 

To  multiply  by  J,  take  a  third  of  the  multipli* 
cand,  that  is,  divide  it  by  3. 

To  multiply  by  f ,  take  J,  first,  and  multiply  that 
by  2 ;  or,  multiply  by  2  first,  and  divide  the  pro- 
duct by  3.* 

^Sometimes  one  operation  is  preferable,  and  soinetimeg 
the  other;  good  judgment  alone  can  decide  when  the 
case  is  before  us. 


70  ORTON's   LIQHTINa   CALCULATOR. 

EXAMPLES. 

1 .  What  will  360  barrels  of  flour  come  to  at  5 J 

dollars  a  barrel.  At  1  dollar  a  barrel  it  would  be  360 

dollars ;  at  5 J  dollars,  it  would  be  SJ  times  as  much, 

360 

5  times,  1800 

J  of  a  time,         90 


Ans.  $1890 

Before  we  attempt  to  divide  by  a  mixed  number, 
such  as  2J,  3J,  5f ,  etc.,  we  must  explain,  or  rather 
observe  the  principle  of  division,  namely:  That 
the  quotient  will  he  the  same  if  we  multiply  the  divi- 
dend and  divisor  hy  the  same  niimher.  Thus  24 
divided  by  8,  gives  three  for  a  quotient.  Now,  if 
we  double  24  and  8,  or  multiply  them  by  any  num- 
ber wnatever,  and  then  divide,  we  shall  still  have  3 
for  a  quotient.     16)48(3;  32)96(3,  etc. 

Now,  suppose  we  have  22  to  be  divided  by  5J ; 
we  may  double  both  these  numbers,  and  thus  be 
clear  of  the  fraction,  and  have  the  same  quotient. 
5J)22(4  is  the  same  as  11)44(4. 

How  many  times  is  IJ  contained  in  12?  An^. 
Just  as  many  times  as  5  is  contained  in  48.  The 
5  is  4  times  1  J,  and  48  is  4  times  12.  From  these 
observations,  we  draw  the  following  rule  for  divid- 
ing by  a  mixed  number. 


MULTIPLICATION   AND   DIVISION.  71 

Rule. — Multiply  the  whole  number  hy  the  lower 
term  of  the  fraction;  add  the  upper  term  to  the  prO' 
duct  for  a  divisor ;  then  multiply  the  dividend  by 
the  lower  term  of  thefraction^  and  then  divide. 

How  many  times  is  li  contained  in  36?  Arts. 
30  times. 

N.  B.  If  we  multiply  both  these  numbers  by  5, 
they  will  have  the  same  relation  as  before,  and  a 
quotient  is  nothing  but  a  relation  between  two 
numbers.  After  multiplication,  the  numbers  may 
be  considered  as  having  the  denomination  of  fifths. 

How  many  times  is  \  contained  in  12  ?  Ans.  48 
times. 

One-fourth  multiplied  by  4,  gives  1 ;  12,  multi- 
plied by  4,  gives  48.  Now,  1  in  48  is  contained  48 
times. 

Divide  132  by  2J.  Ans.  48. 

Divide  121  by  \^.  Am.    8 

How  many  times  is  f  contained  in  3  ?  Ans.  4 
times. 

By  a  little  attention  to  the  relation  of  numbers, 
we  may  often  contract  operations  in  multiplication. 
A  dead  uniformity  of  operation  in  all  cases  indi- 
cates a  mechanical  and  not  a  scientific  knowledge 
of  numbers.  As  a  uniform  principle,  it  is  much 
easier  to  multiply  by  the  small  numbers,  2,  3,  4, 
5,  than  by  7,  8,  9. 


72 


orton's  lightning  calculator. 


Multiply    4532 
by  639 

(63:::z9<7.)    40788 
285516 


Multiply    4532 
by  963 


40788 
285516 


Product,  2895948  Product,  4364316 

In  both  the  foregoing  examples  we  multiply  the 
product  of  9  by  7,  because  7  times  9  are  equal  to  63. 
Because  9  is  in  the  place  of  hundreds  in  exam- 
ple 2,  the  product  for  the  other  two  figures  is  set 
two  places  toward  the  right. 

In  this  last  example  we  may  commence  with  the 
3  units  in  the  usual  way ;  then  that  'product  by  2, 
because  2  times  3  are  6 ;  then  the  product  of  3  by 
3,  which  will  give  the  same  as  the  multiplicand  by 
9.  The  appearance  of  the  work  would  then  be  the 
same  as  by  the  usual  method,  but  would  be  easier, 
as  we  actually  multiply  by  smaller  numbers. 

Multiply    40788 
by  497 


285516 
1998612 

20271636 


Product  of  the  7  units. 
As  7X7=49,  multiply  the 
product  of  7  by  7. 


Every  fact  of  this  kind,  though  extremely  sim- 
ple, should  be  known  by  all  who  seek  for  knowl- 


edge in  figures. 


MULTIPLICATION   AND   DIVISION. 


73 


First  multiply  by  12, 
then  that  product  by  12. 


Multiply 
by 


576 

186 


(6X3    18.) 


3456 
10368 

107136 


Multiply 
by 

Commence  with  6. 
(6X3=18.) 


Multiply  785460 
by  14412 

9425520 
113106240 

11320049520 

Multiply  this  last 
number,  3456,  (which 
is  6  times  576,)  by  3, 
and  place  the  product 
in  the  place  of  tens, 
and  we  have  180  times 
576.  Observe  the  same 
principle  in  the  follow- 
ing examples : 
576  Multiply    40788 

618  by  497 


3456       (7X7=49.)    285516 
10368  1998612 


355968 
Multiply    61524 
by  7209 


20271636 


f^- 


553716 
4429728 


Product,  443646516 


Multiply  this  pro- 
duct of  9  by  8,  because 
9  times  8  are  72,  and 
place  the  product  in 
the  place  of  100,  be- 
cause it  is  7200. 


74         orton's  lightning  calculator. 

Multiply      1243 
by  636 

7458  First  by  600. 

44748        Multiply  7458  by 

Product,        790548 

Multiply    7  8  6  4 
by  24  6 


This  may  be  done  by  commencing  with  the  2; 
then  that  product  by  2  and  3 ;  or  we  may  com- 
mence with  the  6  units,  and  then  that  product  by 
4 ;  because  4  times  6  are  24. 
Multiply  3764  by  199. 

Take  3764  200  times,  and  from  that  product  sub- 
tract 3764. 

Multiply  764  by  498J. 

Take  764  500  times,  and  from  that  product  sub- 
tract 1^  times  764. 

Multiply  396  by  21f ,  or,  (which  is  the  same,) 
99X87=8700  —86    8613. 

N.  B.— Ninety-nine  is  J  of  396,  and  87  is  4 
times  2 If. 

How  many  times  is  125  contained  in  2125  ? 
Same  as  250  in  4250 
Same  as  25  in  425 
Same  as  50  in  850 
Same  as  5  in  85 
Same  as    10  in  170  j  that  is,  17  times. 


MULTIPLICATION   AND   DIVISION.  75 

The  object  of  these  changes  is  to  give  the 
learner  an  accurate  and  complete  knowledg;e  of 
numbers  and  of  division ;  and  the  result  is  not  the 
only  object  sought  for,  as  many  young  learners 
suppose. 

How  many  times  is  75  contained  in  575  ?  or  di- 
vide 575  by  75.  Am.  7f . 

Divide  800  by  12J.  Quotient,  64. 

Divide  27  by  16§.  Quo.  3  j%%,  or  If  J. 

A  person  spent  6  dollars  for  oranges,  at  6 J  cents 
a-piece;  how  many  did  he  purchase?     Ans.  96. 

"When  two  or  more  numbers  are  to  be  multiplied 
together,  and  one  or  more  of  them  having  a  cipher 
on  the  right,  as  24  by  20,  we  may  take  the  cipher 
from  one  number  and  annex  it  to  the  other  with- 
out affecting  the  product;  thus,  24X20  is  the  same 
as  240X2;  286X1300^28600X13;  and  350X 
70x40=35x7x4X1000,  etc. 

Every  fact  of  this  Idnd,  though  extremely  simple, 
will  be  very  useful  to  those  who  wish  to  he  skillful 
in  operation. 

Note. — If  there  are  ciphers  at  the  right  hand 
either  of  the  multiplier  or  multiplicand,  or  of  both, 
they  may  be  neglected  to  the  close  of  the  opera- 
tion, when  they  must  be  annexed  to  the  product. 

Remarks.  —We  now  give  a  few  examples,  for  the  pur- 
pose of  teaching  the  pupil  how  to  use  his  judgment;  he 
will  then  have  learned  a  rule  more  valuable  than  all  others. 
G 


76         orton's  lightning  calculator. 


Multiplication  and  Division   Combined. 

When  it  becomes  necessary  to  multiply  two  or 
more  numbers  together,  and  divide  by  a  third,  or 
by  a  product  of  a  third  and  fourth,  it  must  be  lit- 
erally  done  if  the  numbers  are  prime. 

For  example :  Multiply  19  by  13  and  divide  that 
product  by  7. 

This  must  be  done  at  full  length,  because  the 
numbers  are  prime  ;  and  in  all  such  cases  there  will 
result  a  fraction. 

But  in  actual  business  the  problems  are  almost  all 
reduceable  by  short  operations ;  as  the  prices  of 
articles,  or  amount  called  for,  always  corresponds 
with  some  aliquot  part  of  our  scale  of  computation. 
And  when  two  or  more  of  the  numbers  are  composite 
numberSy  the  work  can  always  be  contracted. 

Example :  Multiply  375  by  7,  and  divide  that 
product  by  21.  To  obtain  the  answer,  it  is  suffi- 
cient to  divide  375  by  3,  which  gives  125. 

The  7  divides  the  21,  and  the  factor  3  remains 
for  a  divisor.  Here  it  becomes  necessary  to  lay 
down  a  plan  of  operation. 

Draw  a  perpendicular  li*ie  and  place  all  numbers 
that  are  to  be  multiplied  together  under  each  other, 
on  the  right  hand  side,  and  all  numbers  that  are 
divisors  under  each  other,  on  the  left  hand  side. 


MULTIPLICATION    AND    DIVISION. 


77 


EXAMPLES. 
Multiply  140  by  36,  and  divide  that  produot  by 
84.     We  place  the  numbers  thus  : 
140 


84 


36 


We  may  cast  out  equal  factors  from  each  side  of 
the  line  without  affecting  the  result.  In  this  case 
12  will  divide  84  and  36  j  then  the  numbers  will 
stand  thus : 

140 


But  7  divides  140,  and  gives  20,  which,  multi- 
plied by  3,  gives  60  for  the  result. 

Multiply  4783  by  39,  and  divide  that  product 
by  13. 

^^     n  3 

Three  times  4783  must  be  the  result. 
Multiply  80  by  9,  that  product  by  21,  and  di- 
vide the  whole  by  the  product  of  60x6X14. 


00 
6 

u 


9 

u 


In  the  above  divide  60  and  80  by  20,  and  14  and 
21  by  7,  and  those  numbers  will  stand  canceled  as 
above,  with  3  and  4,  2  and  3,  at  their  sides.  * 

Now,  the  product  3X^X2,  on  the  divisor  side, 
is  equal  to  4  times  9  on  the  other,  and  the  remain- 
ing 3  is  the  result. 


78         orton's  lightning  calculator. 


General  Rules  for  Cancellation. 

Rule  1st.  Draw  a  perpendicular  line ;  observe 
this  line  represents  the  sign  of  equality.  On  the 
right  hand  side  of  this  line  place  dividends  only, 
on  the  left  hand  side  place  divisors  only ;  having 
placed  dividends  on  the  right  and  divisors  on  the 
left,  as  above  directed. 

2d.  Notice  whether  there  are  ciphers  both  on 
the  right  and  left  of  the  line  j  if  so,  erase  an  equal 
number  from  each  side. 

3d.  Notice  whether  the  same  number  stands 
both  on  the  right  and  left  of  the  line ;  if  so,  erase 
them  both. 

4th.  Notice  again  if  any  number  on  either  side 
of  the  line  will  divide  any  number  on  the  opposite 
side  without  a  remainder ;  if  so,  divide  and  erase 
the  two  numbers,  retaining  the  quotient  figure  on 
the  side  of  the  larger  number. 

5th.  See  if  any  two  numbers,  one  on  each  side, 
can  be  divided  by  any  assumed  number  without  a 
remainder  j  if  so,  divide  them  by  that  number,  and 
retain  only  their  quotients.  Proceed  in  the  same 
manner,  as  far  as  practicable,  then, 

6th.  Multiply  all  the  numbers  remaining  on  the 
rignt  hand  side  of  the  line  for  a  dividend,  and 
those  remaining  on  the  left  for  a  divisor. 

7th    Divide,  and  the  quotient  is  the  answer. 


INTEREST,    DISCOUNT   AND   AVERAGE.  79 

Note. — If  only  one  number  remain  on  either 
side  of  the  line,  that  number  is  the  dividend  or 
divisor,  according  as  it  stands  on  the  right  or  left 
of  the  line  The  figure  1  is  net  regarded  in  the 
operation,  because  it  avails  nothing,  either  to  mul- 
tiply or  divide  by. 

Remarks. — This  method  may  not  work  a  great 
many  problems,  as  they  are  found  in  some  books, 
but  it  will  work  90  out  of  every  100  that  ought  to 
be  found  in  books. 

In  a  book  we  might  find  a  problem  like  this : 

What  is  the  cost  of  21b.  7oz.  13pwt.  of  tea,  at 
7s.  5d.  per  pound.  But  the  person  who  should  go 
to  a  store  and  call  for  31b.  7oz.  and  ISpwt.  of  tea 
would  be  a  fit  subject  for  a  mad-house.  The  above 
problem  requires  downright  drudgery,  which  every 
one  ought  to  be  able  to  perform ;  but  such  drudgery 
never  occurs  in  business. 


interest;  discount,  and  average. 

Before  entering  upon  an  investigation  of  the 
difterent  modes  of  calculating  interest,  it  may  be 
interesting  to  bestow  some  attention  upon  the  his- 
tory of  the  subject,  that  we  may  be  better  prepared 
to  understand  it. 


80  ORTON's   LIQHTNIXG    CALCULATOR. 

Among  the  Jews  a  law  existed  that  they  should 
not  take  interest  of  their  brethren,  though  they 
wero  permitted  to  take  it  of  foreigner**.  "  Thou 
shalt  not  lend  upon  usury  to  thy  brother:  usury 
of  money,  usury  of  victuals,  usury  of  any  thing 
that  is  lent  upon  usury ;  unto  a  stranger  thou  may- 
est  lend  upon  usury ;  but  unto  thy  brother  thou 
Bhalt  not  lend  upon  usury."  (Deuteronomy  xxiii, 
19,  20.)  After  the  dispersion  of  the  Jews  they 
wandered  through  the  earth,  but  they  yet  remain 
a  distinct  people,  mixing,  but  not  becoming  assim- 
ilated with  the  people  among  whom  they  reside. 
Still  looking  to  the  period  when  they  shall  return 
to  the  promised  land,  they  seldom  engage  in  per- 
manent business,  but  pursue  traflBc,  and  especially 
dealing  in  money ;  and  if  their  national  policy  for- 
bids their  taking  interest  of  each  other,  they  show 
no  backwardness  in  taking  it  unsparingly  of  the 
rest  of  mankind.  For  ages  they  have  been  the 
money  lenders  of  Europe,  and  we  may  safely  at- 
tribute to  this  circumstance  the  prejudice,  in  some 
measure,  that  still  exists  even  in  our  own  country 
against  such  as  pursue  this  business  as  a  profession. 
The  prejudice  of  the  Christian  against  the  Jew  has 
been  transferred  to  his  occupation,  and  from  the 
days  of  Shakspeare,  who  painted  the  inexorable 
Bhylock  contending  for  his  pound  of  flesh,  down 
to  the  present  time,  the  grasping  money  lender,  no 


INTERESl,    DISCOUNT,    AND   AVERAGE.  81 

less  than  the  grinding  dealer  in  other  matters,  has 
been  sneeringly  called  a  Jew. 

For  ages  the  taking  of  any  compensation  what- 
ever for  the  use  of  money  was  called  usury,  and 
was  denounced  as  unchristian  ;  and  we  find  Aris- 
totle, the  heathen  philosopher,  gravely  contending 
that  as  money  could  not  beget  money,  it  was  bar- 
ren, and  usury  should  not  be  charged  for  its  use. 
The  philosopher  forgot  that  with  money  the  bor- 
rower could  add  to  his  flocks  and  his  fields,  and 
profit  by  the  produce  of  both. 

Definition  of  Terms. 

Interest  is  premium  paid  for  the  use  of  money, 
goods,  or  property. 

It  is  computed  by  percentage  —  a  certain  per 
cent,  on  the  money  being  paid  for  its  use  for  a 
stated  time.  The  money  on  which  interest  is  paid 
is  called  the  principal. 

The  per  cent,  paid  is  called  the  rate  ;  the  prin- 
cipal and  interest  added   together  is  called   the 

AMOUNT. 

When  a  rate  per  cent,  is  stated,  without  the 
mention  of  any  term  of  time,  the  time  is  under- 
stood to  be  1  year. 

The  first  important  step  in  the  calculation  of 
simple  interest  is  the  arranging  of  the  time  for 
which  it  is  computed.    The  student  must  study  the 


82  ORTON^S   LIGHTNING   CALCULATOR. 

following  Propositions  carefully,  if  Le  would  be 
expert  in  this  important  and  useful  branch  of  bus- 
iness calculations : 

PROPOSITION  1. 
If  the  time  consists  of  years,  multiply  the  principal 
by  the  rate  per  cent.,  and  that  product  by  the 
number  of  years. 

Example  1. — Find  the  interest  of  $75  for  4 
yejvrs  at  6  per  cent. 
Operation. 

$75  The  decimal  for  6  per  cent,  is 

.06  06.     There  being  two  places  of 

decimals  in  the  multiplier,  wo 

4.50  point  oflf  two  in  the  product. 

4 

$18.00  Ans. 

PROPOSITION  2. 
If  the  time  consists  of  years  and  mx)nths,  reduce  the 

time  to  months,  and  multiply  the  principal  by  the 

rate  per  cent,  and  number  of  months  together,  and 

divide  the  result  by  12. 

Note. — The  work  can  always  be  abbreviated  at 
4,  6,  8,  9,  12,  and  15  per  cent,  by  canceling  the 
per  cent.,  or  time,  or  principal,  with  the  common 
d'Tisor  12. 


INTEREST,    DISCOUNT,    AND    AVERAGE,  83 

Example  2.— Find  the  interest  of  $240  for  2 
years  and  7  months  at  8  per  cent. 

First  method.  Second  method  : 

Principal,  $240 

Per  cent.,  .08 


In.  for  lyr.,         19.20 
2yrs.-|-7mos.,  31mns. 


12)595.20 


by  cancellation. 
^40—20 
8  rate. 

31  time. 


n 


49.60  Ans. 


$49.60  ^ns. 

The  operation  by  canceling  is^much  more  brief. 
"We  simply  place  the  principal,  rate,  and  time,  on 
the  right  of  the  line,  and  12  on  the  left ;  then  we 
cancel  12  in  240,  and  the  quotient  20  multiplied 
with  8  and  31  gives  the  interest  at  once. 

Note. — After  12  is  canceled  the  product  of  the 
remaining  numbers  is  always  the  interest. 

PROPOSITION  3. 

Jf  the  time  consists  of  years^  montJis,  and  daySy  re- 
(luce  the  years  to  months^  add  in  the  given  months^ 
and  'place  one-third  of  the  days  to  the  right  of 
this  number y  which  we  multiply  by  the  principal 
and  rate  per  cent.^  and  divide  by  12,  as  before ; 
or  cancel  and  divide  by  12  before  multiplying. 
Example  3. — Find  the  interest  of  $231  for  1 

year,  1  month,  and  6  days,  at  5  per  cent. 


84 


orton's  lightning  calculator. 


First  method. 
Principal, 
Per  cent., 


$231 
.05 


In.  for  lyr.,  11.55 

lyr.-f  lmo.+6da.,    13.2mo. 

12)152.460 


Second  method: 
by  cancellation. 
231  prin. 

5  rate. 

m—n 


n 


$12,705  Am. 


$12,705  An%. 

By  the  second  method  we  cancel  12  in  132,  and 
multiply  the  quotient  11  by  5  and  231. 

Note. — When  the  principal  is  $,  and  the  time  is 
in  years  or  months,  the  interest  is  in  cents ;  if  the 
time  is  in  years,  months,  and  days,  the  interest  is 
in  mills,  unless  the  days  are  less  than  3,  in  which 
case  it  would  be  in  cents,  as  before. 

Note. — The  reason  we  divide  the  days  by  3  is 
because  we  calculate  30  days  for  a  month,  and  di- 
viding by  3  reduces  the  days  to  the  tenth  of  months. 

Note. — The  three  preceding  propositions  will 
work  any  note  in  interest  for  any  time  and  at  any 
given  rate  per  cent. 

Mow  to  Avoid  Fractions  in  Interest. 

PROPOSITION  4. 

If^  when  the  time  consists  of  years^  months^  and 

days,  are  not  divisible  by  3,  you  can  divide  the  days 

hy  3,  and  annex  the  mixed  number  as  in  Proposition 


INTEREST,  DISCOUNT,  AND  AVERAOE. 


85 


3;  or  if  you  wish  to  avoid  fractioTis^  you  can  reduce 
the  time  to  interest  days^  and  multiply  the  principal^ 
rate  and  days  togethery  and  divide  the  result  by  36 
or  its  factors^  4X  9. 

Note. — la  this  case  as  in  the  preceding,  the 
work  can  almost  always  be  contracted  by  dividing 
the  rate  or  time  or  principal  with  the  divisor  36. 

Note. — We  use  the  divisor  36,  because  we  cal- 
culate 360  interest  days  to  the  year.  We  discard 
the  0,  because  it  avails  nothing  to  multiply  or  di- 
vide by. 

Example  4. — !Pind  the  interest  of  $210  for  1 
year,  4  months,  and  8  days,  at  9  per  cent. 

Year.        Months.        Days. 


1  4 

Operation 

By  Prop.  3. 

$210 
.9 

18.90 
16.2§ 

12)307440 


8=16.21  months  or  488  days. 

Operation 

By  Prop.  4. 

$210 
9 

18.90 

488 


$25,620  Ans, 


36)922320 


$25,620  ^rw. 


We  will  now  work  the  example  by  cancellation 
to  show  its  brevity. 


86  orton's  lighting  calculatoe. 

Operation  hy  Cancellation, 
Time  488  days. 

210 

m  122 

122 
210 


325.620 

Now  cancel  9  in  36  goes  4  times,  then  4  into  488 
goes  122.  Now  multiply  remaining  numbers  to- 
gether, thus,  210X-22  and  we  have  the  interest  at 
once. 

When  the  days  are  not  divisible  by  3  we  reduce 
the  whole  time  to  days ;  then  we  place  the  princi- 
pal rate  and  time  on  the  right  of  the  line.  Now, 
because  the  time  is  in  days,  we  place  36,  on  the 
left  of  the  line  for  a  divisor.  (Jf  the  time  was 
months  we  would  place  12  on  the  left.") 

Note. — A  very  short  method  of  reducing  time 
to  interest  days  is  to  multiply  the  years  by  36 ; 
add  in  3  times  the  number  of  months  and  the  tens' 
figure  of  the  days,  and  annex  the  unit  figure;  but 
if  the  days  are  less  than  10  simply  annex  them. 

Example  1. — Reduce  1  year,  2  months,  and  6 
days,  to  days. 

Tears.        Months.  Days. 

36Xl-j-''^X2==42  annex  6=426  Am. 


SIMPLE  INTEREST  Br  CANCELLATION.  87 

Example  2. — Reduce  2  years,  3  montlia    and 
17  days,  to  interest  days. 

Years.     M'ths.  Days.  Days. 

36x2-1-3x34-1=82.  annex  7=827  days  Aiu, 
Note. — The  student  should  commit  to  mei*:ory 
the  multiplication  of  the  number  36  up  as  far  as 
9  times   36,  and  then  he  can  reduoe  almost  in- 
stantly years,  months,  and  days,  to  days. 


SIMPLE  INTEREST  BY  CANCELLATION, 

Rule. — Place  tJie  principal,  time,  and  rate  per 
cent,  on  the  right  hand  side  of  the  line.  If  the  time 
consists  of  years  and  months,  reduce  them  to  months^ 
and  place  12  (the  number  of  months  in  a  year')  on 
the  left  hand  side  of  the  line.  Should  the  iim^  con- 
iist  of  months  and  days,  reduce  them  to  days  or  deci- 
mal parts  of  a  month.  If  reduced  to  days,  place 
36  071  the  left.  If  to  decimals  parts  of  a  Trwnthy 
place  12  only  as  before. 

Point  off  two  decimal  places  when  the  time  is  in 
months,  and  three  decimal  places  when  the  time  is 
in  days. 

Note.  If  the  principal  contains  cents,  point  oflf 
four  decimal  places  when  the  time  is  in  months, 
and  five  decimal  places  when  the  time  is  in  days. 
H 


88  OETON'S   LiaHTNINQ   CALCULAIOR. 

^OTE. —  We  place  36  on  the  left  because  there  ^^ 
360  interest  days  in  a  year.  (^Custom  has  made  this 
lawful.^ 

Example  1. — ^What  is  the  interest  on  $60  for 
117  days  at  6  per  cent? 
Operation. 

Here  117X0 
must  be  the  $0 
answer. 


00  Both  sixes    on    tho 

0  right  cancels  36  on 

117  the    left,    and    we 

have    nothing    left 


$1,170  Ans.     to  divide  by. 
In  this  case  we  point  off  three  decimal  places  be- 
cause the  time  is  in  days.    If  the  time  had  been  117 
months,  we  would  have  pointed  off  but  two  deci- 
mal places. 

Example  2. — What  is  the  interest  of  $96.50 
for  90  days  at  6  per  cent? 
Operation. 

96.50  9650 

0—^0     00—15  15 

0  

1.44.750  Ans. 

Now  cancel  6  in  36  and  the  quotient  6  into 
90,  and  we  have  no  divisor  left.  Hence  15X96.50 
must  be  the  answer. 

Note — As  there  are  cents  in  the  principal,  we 
point  off  five  decimals  ;  three  for  days  and  two  for 
cents.  Pay  no  attention  to  the  decimal  point  tintiJ 
the  close  of  the  operation. 


SIMPLE    INTEREST   BY   CANCELLATION. 


89 


Example  3.— What   is  the   interest  of  $480 

for  361  days  at  6  per  cent? 

^^0—80                      361 

^-H 

361                                 80 

0 

$28,880  Ans. 

Now  cancel  6  in  36  and  the  quotient  6  into  480, 

and  we  have  no  divisor  left.     Hence  80X361  must 

be  the  answer. 

Example  4.— What  is  the  interest  of  $720  for 

9  months  at  7  per  cent? 

n0—eo                60 

1^ 

9                                         9 

7                                  

540 
7 

$37.80  Ans. 

Now  cancel  12  in  720  there  is  nothing  left  to 
divide  by.     Hence  60x9x7  must  be  the  answer. 

N.  B.  When  interest  is  required  on  any  sum  for 
days  only,  it  is  a  universal  custom  to  consider  30 
days  a  month,  and  12  months  a  year ;  and,  as  the 
unit  of  time  is  a  year,  the  interest  of  any  sum  for 
one  day  is  g^^,  what  it  would  be  for  a  year.  For 
2  days,  3|q,  etc.;  hence  if  we  multiply  by  the 
days,  we  must  divide  by  360,  or  divide  by  36  and 
save  labor.  The  old  form  of  this  method  was  to 
place  360,  or  12  and  30,  on  the  left  of  the  line, 
but  using  36  is  much  shorter. 


90  ORTONS   LIGHTNING   CALCULATOR. 

WHEN   THE   DAYS   ARE  NOT  DIVISIBLE   BY  THREE. 

Note. — When  the  time  consists  of  months  and 
days,  and  the  days  are  not  divisible  by  three,  re- 
dace  the  time  to  days. 

Example  5. — What  is  the  interest  of  $960  for 
1 1  months  and  20  days  at  6  per  cent? 

Months.     Days. 

Operation.  11     20=350  days. 

000—160  350 

0  —36     350  160 

6  

$56,000 
Now   cancel  6  in  36  and  the  quotient  6  into 
960,  and  we  have  no  divisor  left.     Hence  160X 
350  must  be  the  answer. 

Example  6. — What  is  the  interest  of  $173  for 
8  months  and  16  days  at  9  per  cent? 

Months.     Days. 

Operation.  8     16=256  days. 

173  173 

4r-n     0  64 

^00^64  

$11,072  Ans. 
Now  cancel  9  in  36  and  the  quotient  4  into  256, 
and  we  have  no  divisor  left.    Hence  64X173  must 
be  the  answer. 

N.  B.  Let  the  puj  il  remember  that  this  is  a  gen- 
eral and  universal  method,  equally  applicable  to 
any  per  cent,  or  any  required  time,  and  all  other 
rules  must  be  reconcilable  to  it;  and,  in  fact,  all 
other  rules  are  but  modifications  of  this. 


SIMPLE   INTEREST    BY    CANCELLATION.         91 

Example  7. — What  is  the  interest  on  $1080 
for  7  months  and  11  days  at  7  per  cent? 

Months.    Days. 

7        11=221  days. 

Operation. 

^0^0—30  221 

221  30 

7  

6630 

7 


H 


Now 

$46,410  Ans. 
cancel  36  in  1080  and  we  have  no  divisor 

lefl,  hence  30X221X7  must  be  the  answer. 

WITH  MORE  DIFFICULT  TIME  AND  RATE  PER  CENT. 

Example  8.— What  is  the  interest  of  $160  for 

19  months  and  23  days  at4J  per  cent? 

Opera 

Months.    Days. 

19         23=593  days, 
tion. 
160—20                                    593 
593                                                20 

H                                      

$11,860  A71S. 
Now  cancel  4J  in  36  and  the  quotient  8  into  160 
we  have  no  divisor  left,  hence  20x593  must  be 
the  interest. 

WHEN    THE   DAYS    ARE    DIVISIBLE    BY   THREE. 

Kule. — Place  one-third  of  the  days  to  the  rigid 
'if  the  months,  and  place  12  on  the  left  of  the  line. 


92 


ORTCN  S    LIGHTNING   CALCULATOR. 


Example  11. — What  is  the  interest  of  $350  for 
3  years  7  months  and  6  days  at  10  per  cent? 


Years.    Months.    Days. 


Operation. 
350 


It 


10 


6=43.2  months. 

350 
36 


12600 
10 


$126,000  Am, 

Now  cancel  12  in  432  and  \fe  have  no  divisor 
left.     Hence  350X36X10  equals  the  interest. 

Example  12.— What  is  the  interest  of  $241  for 
13  months  and  9  days  at  8  per  cent? 

Months.    Days. 

13         9=13.  3  months. 


Operation. 
241 


z-n 


13.3 


241 
133 


32053 
2 


3)64106 


$21.368f^«5. 
In  this  example  I  canceled  8  and  12  by  4,  and 
then  multiplied  all  on  the  right  of  the  line  and  di- 


SIMPLE  INTEREST  BY  CANCELLATIO.V.    93 

vided  by  3.  If  I  could  have  divided  by  3  before 
multiplying  I  would  have  saved  labor,  but  when  the 
numbers  are  prime  the  whole  work  must  be  liter' 
ally  done. 

Closing  Remarks. — We  have  now  fully  ex- 
plained the  canceling  system  of  computing  inter- 
est. Any  and  every  problem  can  be  stated  by  this 
method,  and  the  beauty  and  simplicity  of  the 
system  ranks  it  high  among  the  most  important 
abbreviations  ever  discovered  by  man.  As  we  have 
before  remarked,  at  6,  4,  8,  9,  12,  15,  and  4^  per 
cents.,  every  problem  in  interest  can  be  canceled, 
besides  a  great  many  can  be  abbreviated  at  5,  7,  and 
other  per  cents.;  and  after  the  problem  has  been 
stated  and  we  find  that  we  can  not  cancel,  what 
have  we  done?  We  have  simply  stated  the  prob- 
lem in  its  simplest  and  easiest  form  for  working  it 
by  any  other  method.  Hence  we  have  a  decided 
advantage  of  all  notes  that  will  cancel,  and  if  we 
can  not  cancel  we  have  stated  the  problem  in  its 
correct  and  proper  form  for  going  through  the 
whole  work ;  but  it  is  only  when  the  principal, 
time,  and  rate  per  cent,  are  all  prime,  that  the 
WHOLE  work  must  be  literally  done.  At  6  per 
cent,  we  can  cancel  through,  and  6  is  the  rate  most 
commonly  used 


91         orton's  lightning  calculator. 


SHORT    PRACTICAL    RULES, 

DEDUCED  FROM  THE  CANCELING  SYSTEM, 

For  calculating  interest  at  6  per  cent.j  either  for 
months,  or  months  and  days. 

To  find  the  interest  for  months  at  6  per  cent. 

Rule. — Multiply  the  principal  hy  half  the  num- 
ber of  months,  expressed  decimally  as  a  per  cent., 
that  is,  for  12  months,  multiply  hy  .06  ;  for  8  months, 
multiply  hy  .04. 

Note  1. — It  is  obvious  that  if  the  rate  per  cent, 
were  12,  it  would  be  1  per  cent,  a  month  ;  if,  there- 
fore, it  be  6  per  cent.,  it  will  be  a  half  per  cent,  a 
month  J  that  is,  half  the  months  will  be  the  per 
cent. 

Note  2. — If  any  other  per  cent.  U  wanted  you 
can  proceed  as  above,  and  then  multiply  by  the 
given  rate  per  cent,  and  divide  by  6,  and  the  quo- 
tient is  the  interest. 

1.  WhLt  is  the  interest  of  $368  for  8  months? 

$368 
.04=half  the  months. 


%14:.72=Ans. 
Note  3. — When  the  months  are  not  even;  that  is, 
will  not  divide  by  2,  multiply  one-half  the  principal 


SHORT  PRACTICAL  RULES.         95 

by  tlie  whole  number  of  months,  expressed  deci* 
mally. 

To  find  the  interest  of  any  sum  at  6  per  cent, 
per  annum  for  any  number  of  months  and  days. 

Rule. — Divide  the  days  hy  3  and  place  the  quo- 
lient  to  the  right  of  the  months;  one-half  of  the  nwm- 
her  thus  formed  multiplied  hy  the  principal^  or  one- 
half  of  the  principal  multiplied  hy  this  number  J  will 
give  tlie  interest — 'pointing  off  three  decimal  places 
when  the  principal  is  $. 

2.  What  is  the  interest  of  $76  for  1  year.  6 
months,  and  12  days,  at  6  per  cent? 

Years.  Months.  Days. 

1        6        12=18.4months— half  9  2. 
$76  Or,  184 

9.2  38=rhalf  prin. 


$6,992  Ans.  $6,992  Ans. 

Note. — Dividing  the  days  by  3  reduces  them  to 
the  tenth  of  months. 

To  find  the  the  interest  of  any  sum  at  6  per 
cent,  per  annum  for  any  number  of  days. 

Rule. — Divide  the  principal  hy  6  aiid  mtdtiply 
the  quotient  hy  the  number  of  days;  or  divide  the 
days  by  6  and  multiply  the  quotient  hy  the  principal^ 
pointing  off  three  decimal  places  when  the  principal 
is  $. 

Note. — Always  divide  6  into  the  number  that 


96         orton's  lightning  calculator. 

will  divide  without  a  remainder;  if  neither  one 
will  divide,  multiply  the  principal  and  days  to- 
gether and  divide  the  result  by  6. 

3.  What  is  the  interest  of  S240  for  18  days  at 
6  per  cent? 

18h-6=3  240-f-6=40 

$240  Or,  $40=J  of  prin. 

3=:J  of  the  days.  18 

$0,720  ^715.  $0,720  Ans. 

4.  What  is  the  interest  of  $1800  for  72  days  at 
6  per  cent. 

$1800  Or,  $300=J  of  prin. 

12=rJ  of  the  days.  72 


$21,600  Ans.  $21,600  Ans. 

Useful  Suggestions  to  the  Accountant  in  Computing 
Interest  at  Q  'per  cent. 

If  the  principal  is  divisible  by  6,  always  reduce 
the  time  to  days;  then  multiply  the  number  of 
days  by  one -sixth  of  the  principal. 

EXAMPLE. 
5.  Find  the  interest  of  $240  for  1  year,  5  months, 
and  17  days,  at  6  per  cent. 

6)240  lyr,  5raos.,  17da.=:527  days. 

Multiplied  by    40 

J  of  prin.=40  

$21,080  Ans. 


SHORT   PRACTICAL    RULES.  97 

If  the  days  are  only  divisible  by  3,  multiply 
one-third  of  the  principal  by  one-half  of  the  days, 

6.  What  is  the  interest  of  $210  for  80  days  at  6 
per  cent.  ? 

$70=J  of  the  principal. 
40=1  of  the  days. 

$2,800  Ans. 

When  the  Rate  of  Interest  is  4  per  cent 
Rule. — Multiply  the  principal  hy  one-third  the 
number  of  months,  or  hy  one-ninth  the  number  of 
days,  and  the  product  is  the  interest. 

Note. — This  principle  is  also  deduced  from  the 
canceling  method  of  computing  interest ;  the  stu- 
dent can  readily  see  that  4  is  J  of  12  and  ^  of  36, 

When  the  Bate  of  Interest  is  9  per  cent 
KuLE. — Multiply  the  principal  by  three-fourths 
the  number  of  months,  or  one-fourth  the  number  of 
days,  or  vice  versa. 


BANKS  AND  BANKING. 


A  bank  is  an  institution  or  corporation  for  the 
purpose  of  trafficking  in  money. 

Banks  receive  money  on  deposit,  loan  money  on 
interest,  and  issue  bank-notes,  i.  e.,  notes  payable 
in  specie  to  the  bearer  on  demand,  and  which  cir- 
culate as  money. 

A  promissory  note,  nfore  commonly  called  a  notej 
is  a  written  promise  to  pay  a  specified  sum  of  money. 

A  person  who  indorses  a  note  incurs  all  the  obli- 
gations of  such  an  indorsement,  even  though  he 
may  be  ignorant  of  them  at  the  time.* 

*  Many  a  man  has  been  reduced  from  affluence  to  poverty,  by 
merely  writing  his  name  on  the  back  of  a  note  "just  to  accom- 
modate a  friend." 

98 


BANKERS    METHOD  CP  COMPUTING  INTEREST.  99 

BANKERS'   METHOD 
or 

COMPUTING    INTEREST, 

AT  6  PER  CENT.  FOR  ANT  NUMBER  OP  DAYS. 


Rule. — Draw  a  perpendicular  UnCj  cutting  off 
the  two  right  hand  figures  of  the  $,  and  you  have  the 
interest  of  the  sum  for  60  days  at  Qper  cent. 

Note. — The  figures  on  the  left  of  the  line  arc  S, 
and  those  on  the  right  are  decimals  of  $. 

Example  1. — What  is  the  interest  of  8423 
60  days  at  6  per  cent  ? 

$423=the  principal. 

$4  I  23  cts.=:interest  for  60  days. 

Note. — When  the  time  is  more  or  less  than  60 
days,  first  get  the  interest  for  60  days,  and  from 
that  to  the  time  required. 

Example  2. — What  is  the  interest  of  $1 24  for 
15  days  at  6  per  cent.? 

Days.  Days, 

15=i  of  60 

$124:=principal. 

4)1  I  24  cts.=interest  for  60  days. 

I  31  cts.=rintercst  for  15  days. 
I 


100  orton's  lightning  calculator. 

Example  3. — What  is  the  interest  of  $123.40  for 
90  days  at  6  per  cent.? 

Days.  Days.  Days. 

90=60-L30 
$123.40=principal. 


2)1 


2340=interest  for  60  days. 
6170=interest  for  30  days. 


An!f.   U  I  851z=interest  for  90  days. 

Example  4. — AVhat  is  the  interest  of  $324  for 
75  days  at  6  per  cent.? 

Days.  Days.  Days. 

$324=:principal.  75=60+ 1 5 


4)3 


24  cts.  interest  for  60  days. 
81  cts.  interest  for  15  days. 


A?is.  $4  I  05  cts.  interest  for  75  days. 

Kemarks. — This  system  of  Computing  Interest 
is  very  easy  and  simple,  especially  when  the  days 
are  aliquot  parts  of  60,  and  one  simple  division 
will  suffice.  It  is  used  extensively  by  a  large  ma- 
jority of  bur  most  prominent  bankers ;  and,  indeed, 
is  taught  by  most  all  Commercial  Colleges  as  the 
shortest  system  of  computing  interest. 

Method  of  Calculating  at  Different  Per  Cents. 

This  principle  is  not  confined  alone  to  6  percent, 
as  many  suppose  who  teach  and  use  it.  It  is  their 
custom  first  to  find  the  interest  at  6  per  cent.,  and 
from  that  to  other  per  cents.  But  it  is  equally  ap- 
plicable foi  a?Z  per  cents.,  from  1  to  15  inclusive. 


bankers'  method  of  computing  interest.  101 

The  following  table  shows  the  diiFerent  per  cents., 
with  the  time  that  a  given  number  of*  $  will 
amount  to  the  same  number  of  cents  when  placed 
at  interest. 

Rule. — Draw  a  perpendicular  line,  cutting  off 
tlie  two  right  hand  figures  of  $,  and  you  have  the 
interest  at  the  following  percents. 

Interest  at  4  per  cent,  for  90  days. 

Interest  at  5  per  cent,  for  72  days. 

Interest  at  6  per  cent,  for  60  days. 

Interest  at  7  per  cent,  for  52  days. 

Interest  at  8  per  cent,  for  45  days. 

Interest  at  9  per  cent,  for  40  days. 

Interest  at  10  per  cent,  for  36  days. 

Interest  at  12  per  cent,  for  30  days. 

Interest  at  7-30  per  cent,  for  50  days. 

Interest  at  5-20  per  cent,  for  70  days. 

Interest  at  10-40  per  cent,  for  35  days. 

Interest  at  7 J  per  cent,  for  48  days. 

Interest  at  4J  per  cent,  for  80  days. 
Note. — The  figures  on  the  left  of  the  perpen- 
dicular line  are  dollars,  and  on  the  right  decimals 
of  $.     If  the  $  are  less  than  10  prefix  a  0. 

Example  1. — What  is  the  interest  of  ^120  for 
15  days  at  4  per  cent? 

Days.  Havs. 

$120=:principal.  15= J  of  90. 

6)1  I  20  cts.-int  for  90  days. 
I  20  cts.— int.  for  15  days. 


102       orton's  lightning  calculator. 

Example  2.— What  is  the  interest  of  $132  for 
13  days  at  7  per  cent.  ? 

Days.  Days. 

$132=principal.  13=J  of  52. 

4)1     32  cts.=rint.  for  52  days. 
33  cts.rziint.  for  13  days. 
Example  3. — What  is  the  interest  of  $520  for 
9  days  at  8  per  cent.  ? 

Days.  Days. 

$520=principal.  9=^  of  45. 

5)5  I  20  cts.=int.  for  45  days. 
$1  I  04  cts.=int.  for  9  days. 

Example  4. — What  is  the  interest  of  $462  for 
for  64  days  at  7 J  per  cent.  ? 

Days.  Days.  Days. 

$462=principal.  64=4^+16. 


3)4 

1 


62  cts.  —int.  for  48  days. 
54  cts.  ir^int.  for  16  days. 


$6  I  16  cts.=int.  for  64  days. 

Remark. — We  have  now  illustrated  several  ex- 
amples by  the  different  per  cents. ;  and  if  the  stu- 
dent will  study  carefully  the  solution  to  the  above 
examples,  he  will  in  a  short  time  be  very  rapid  in 
this  mode  of  computing  interest. 

Note. — The  preceding  mode  of  computing  in- 
terest is  derived  and  deduced  from  the  canceling 
system ;  as  the  ingenious  student  will  readily  see. 
It  is  a  short  and  easy  way  of  finding  interest  for 
days  when  the  days  are  even  or  aliquot  parts ;  but 
when  they  are  not  multiples,  and  three  or  four  dV 


bankers'  method  op  computing  interest.  103 

visions  are  ncessary,  the  canceling  system  is  much 
more  simple  and  easy.  We  will  here  illustrate  an 
example  to  show  the  diflference :  Required  the  in- 
terest of  $420  for  49  days  at  6  per  cent. 


2)4 

2)2 
5)1 
3) 


Bankers'  method. 

20  cts.=int.  for  60  days. 

10  cts.=rint.  for  30  days. 
05  cts.=int.  for  15  days. 

21  cts.=int.  for  3  days. 
7  cts.^int.  for  1  day. 

Canceling  moth. 
^^0—70 

^-H       0 
49 
70 

$3,430  Ans. 

$3  I  43  cts.=int.  for  49  days. 

The  canceling  method  is  much  more  brief;  we 
simply  cancel  6  in  36,  and  the  quotient  G  into  420  ; 
there  is  no  divisor  left;  hence  70X49  gives  the  in- 
terest at  once. 

If  the  time  had  been  15  or  20  days,  the  Bankers' 
Method  would  have  been  equally  as  short,  because 
15  and  20  are  aliquot  parts  of  60.  The  superiority 
the  canceling  system  has  above  all  others  is  this :  it 
takes  advantage  of  the  principal  as  well  as  the  time. 

For  the  benefit  of  the  student,  and  for  the  con- 
venience of  business  men,  wo  will  investigate  this 
system  to  its  full  extent  and  explain  how  to  take 
advantage  of  the  jirincipal  when  no  advantage  can 
be  taken  of  the  dai/s.  This  is  one  of  the  most  im- 
portant characteristics  of  interest,  and  very  oftea 
saves  much  labor.  It  sliould  he  used  when  the  dayt 
are  not  even  or  aliquot  parts. 
I* 


104  ORTON'S    LIGHTNING    CALCULATOR. 

The  following  table  shows  the  different  sums  of 
money  (at  the  different  per  cents.)  that  bear  1  cent 
interest  a  day ;  hence  the  time  in  days  is  always 
the  interest  in  cents ;  therefore,  to  find  the  interest 
on  any  of  the  following  notes  at  the  per  cent,  at- 
tached to  it  in  the  table,  we  have  the  following 
rule : 

Rule. — Draw  a  perpendicular  line,  cutting  off 
tJie  two  right  hand  figures  of  the  days  for  centSj  arid 
you  have  the  interest  for  the  given  time. 

Interest  of  $90  at  4  per  cent,  for  1  day  is  1  cent. 

Interest  of  $72  at  5  per  cent,  for  1  day  is  1  cent. 

Interest  of  $60  at  6  per  cent,  for  1  day  is  1  cent. 

Interest  of  $52  at  7  per  cent,  for  1  day  is  1  cent. 

Interest  of  $45  at  8  per  cent,  for  1  day  is  1  cent. 

Interest  of  $40  at  9  per  cent,  for  1  day  is  1  cent. 

Interest  of  $36  at  10  per  cent,  for  1  day  is  1  cent. 

Interest  of  $30  at  12  per  cent,  for  1  day  is  1  cent. 

Interest  of  $50  at  7.30  per  ct.  for  1  day  is  1  ct. 

Interest  of  $70  at  5.20  per  ct.  for  1  day  is  1  ct. 

Interest  of  $35  at  10.40  per  ct.  for  1  day  is  1  ct. 

Interest  of  $48  at  7  J  per.  cent,  for  1  day  is  1  cent. 

Interest  of  $80  at  4J  per  cent,  for  1  day  is  1  cent. 

Interest  of  $24  at  15  per  ct.  for  1  day  is  1  cent. 

Note. — The  7.30  Government  Bonds  are  calcu- 
lated on  the  base  of  365  days  to  the  year,  amd  the 
5.20's  and  10.40's  on  the  base  of  364  days  to  the  year. 


BANKERS   METHOD  OF  COJirUTirfO  INTEREST.    105 

Note. — This  table  should  be  committed  to  mem- 
01  y,  as  it  is  very  useful  when  the  days  are  not  even 
or  aliquot  parts.  If  the  days  are  less  than  10  pre- 
fix a  0  before  drawing  the  line. 

Example  1. — Required  the  interest  of  $60  for 
117  days  at  6  per  cent. 

llT^the  days.  Here  we  cut  oflF  the  two 

$1  I  17  cts.  Ans.       right  hand  figures  for  cents. 

The  student  should  bear  in  mind  that  the  inter- 
est on  $60  for  117  days  is  just  the  same  as  the 
interest  on  $117  for  60  days. 

By  looking  at  the  table  we  see  that  the  interest 
for  $60  at  6  per  cent,  is  1  cent  a  day ;  hence  the 
time  in  days  is  the  answer  in  cents.  If  this  note 
was  $120,  instead  of  $60,  we  would  first  find  the 
interest  for  $60,  and  then  double  it;  if  it  was 
$180,  we  would  multiply  by  3,  etc. 

Example  2. — Required  the  interest  of  $45  for 
219  days  at  8  per  cent. 

219=the  days.  Here  we  cut  off  the  two 

$2  I  19  cts.  Ans.       right  hand  figures  for  cents. 

The  student  should  bear  in  mind  that  the  inter- 
est on  $45  for  219  days  is  just  the  same  as  the 
interest  on  $219  for  45  days. 

By  looking  at  the  table  we  see  that  the  interest 
on  $45  at  8  per  cent,  is  1  cent  a  day ;  hence  the 
time  in  days  is  the  answer  in  cents.     If  this  note 


106       orton's  lightning  calculator. 

was  $22.50,  instead  of  $45,  we  would  first  get  the 
interest  for  $45,  and  then  divide  by  2 ;  if  it  was 
875,  we  would  add  on  f ;  if  $60,  add  on  i  etc. 

Example  3.— Required  the  interest  of  $48  for 
115  days  at  9  per  cent. 

115— the  days.  $48=$40-|-$8. 

5)$1  I  15  cts.z=the  int.  of  $40  for  115  days. 
I  23  cts.=rthe  int.  of  $8  for  115  days. 

Ans.  $1  I  38  cts.=the  int.  of  $48  for  115  days. 

Here  we  first  find  the  interest  of  $40,  because 
the  days  is  the  interest  in  cents;  then  we  divide  by 
5  to  find  the  interest  for  $8 ;  then  by  adding  both 
we  find  the  interest  for  $48,  as  required. 

Example.  4 — Required  the  interest  of  $260  for 
104  days  at  7  per  cent. 

$52X5=:$260. 
104=the  days. 

$1     04  ctszzzthe  int.  of  $52  for  104  days. 
A71S.  $5     20  cts.     Multiply  by  5. 

Here  we  first  find  the  interest  of  $52,  because 
the  days  is  the  interest  in  cents ;  then  we  multiply 
by  5  to  get  it  for  $260.  We  could  have  worked 
this  note  by  the  Bankers'  Method,  just  as  well,  by 
cutting  off  two  figures  in  the  principal,  making 
$2.60  cts.  the  interest  for  52  days,  and  then  multi- 
ply by  2  to  get  it  for  104  days.  %\e  student  must 
remember  that  the  interest  of  $260  for  104  days  is 
just  the  same  as  the  interest  of  $104  for  260  days. 


bankers'  aiETnoi)  of  computing  interest.  107 

Prohlems  Solved  hy  Both  Mctlwds. 
"We  will  now  solve  some  examples  by  both  metli- 
ods,  to  further  illustrate  this  system,  and  for  the 
purpose  of  teaching  the  pupil  how  to  use  his  judg- 
ment. He  will  then  have  learned  a  rule  more  val- 
uable than  all  others. 

Example  5. — What  is  the  interest  S180  for  75 
days  at  6  per  cent.? 

Operation  by  taking  advantage  of  the  $. 
75=the  days.  $60x3r=S180, 

$0  I  75  cts.=:the  int.  of  $60  for  75  days. 
I    3  Multiply  by  3. 


Ans.    §2  I  25  cts.=the  int.  of  $180  for  75  days. 
Operation  by  the  Bankers'  Method. 
6180=the  principal.  60da.-f  15da.=75da. 


4)§1 


80  cts.=the  int.  for  60  days. 
45  cts.=the  int.  for  15  days. 


An$.   $2  I  25  cts.r^the  int.  for  75  days. 

By  the  first  method  we  multiplied  by  3,  because 
3X160=^180;  by  the  second  method  we  added 
on  J-,  because  60da.-[-'V°da.=:75da, 

N.  B, — When  advantage  can  be  taken  of  both 
time  and  principal,  if  the  student  wishes  to  prove 
his  work,  he  can  first  work  it  by  the  Bankers* 
Method,  and  then  by  taking  advantage  of  the  prin- 
cipal, or  vice  versa.  And  as  the  two  operations  are 
entirely  difierent,  if  the  same  result  is  obtained  by 
each,  he  may  fairly  conclude  that  the  work  is  correct. 


108  ORTON'S    LIGHTNINa    JALCULATOB. 

LIGHTNING  METHOD 

OP 

COMPUTING  INTEREST 

On  all  notes  that  hear  $12  j)er  annum,  or  any  ali^ 
quotpart  or  multiple  of  $12. 

If  a  note  bears  $12  per  annum,  it  will  certainly 
bear  $1  per  month ;  hence  the  time  in  months 
would  be  the  interest  in  $ ;  and  the  decimal  parts 
of  a  month  would  be  the  interest  in  decimal 
part«  of  a  $;  therefore  when  the  note  bears  $12 
per  annum  we  have  the  following  rule : 

Rule. — Reduce  tlie  years  to  months,  add  in  the 
given  months,  and  place  one-third  of  the  days  to  the 
right  of  this  number,  and  you  have  the  interest  in 
dimes. 

Example  1. — Required  the  interest  of  $200 
for  3  years,  7  months,  and  12  days,  at  6  per  cent. 

200  J  of  12  days=4. 

6 


T>.  Mo.  Da. 


$12.00=:int.  for  I  yr.  3     7  12rrr43.4mo. 

Hence  43.4  dimes,  or  $43.40cts.,  Ans. 

"We  see  by  inspection  that  this  note  bears  $12 
interest  a  year;  hence  the  time  reduced  to  mouths, 


HGQTNING  METHOD  OF  COMPUTING  INT,     109 

with  one-third  of  the  days  to  the  right,  is  the  in- 
terest in  dimes.  If  this  note  bore  $6  a  year,  in- 
stead of  $12,  we  would  take  one-half  of  the  above 
interest;  if  it  bore  $18,  instead  of  $12,  we  would 
add  one-half;  if  it  bore  $24,  instead  of  $12,  we 
would  multiply  by  2,  etc. 

Example  2. — Required    the   interest   of  $150 
for  2  years,  5  months,  and  13  days,  at  8  per  cent. 
150  J  of  13  days=:4J. 


8 


Yr.  Mo.  r>a. 


$12.00r=int.  for  1  yr.  2    5  13=:r29.4Jmos. 

Hence  $29.4J  dimes,  or  $29.43J  cts.,  A7is. 
We  see  by  inspection  that  this  note  bears  $12f 
interest  a  year;  hence  the  time  reduced  to  months, 
with  one-third  of  the  days  placed  to  the  right,  gives 
the  interest  at  once. 

Example  3. — Required  the  interest  of  $160  for 

11  years,  11  months,  and  11  days,  at  7J  per  cent. 

IGO  J  of  11  days=3|. 

"  Tr.  Mo,  Da 

$]2.00=:int.forlyr.  11  11  ll=143.3|mos. 

Hence  $143.3J  dimes,  or  $143.3(;f  cts.,  Ans. 

When  the  Interest  is  more  or  less  than  $12  a  Year. 
Rule. — First  find  the  interest  for  the  given  time 
on  the  hase  o/  $12  interest  a  year;  then,  if  the  in- 
terest on  the  note  is  only  $G  a  year^  divide  by  2  ;   if 


110  ORTON's   LIGHTNINa   CALCULATOR 

$24  a  year,  multiply  ly  2j  if  $18  a  year,  add  on 
one-half,  etc. 

Example  1. — What  is  the  interest  of  $300  for 
4  years,  7  months,  and  18  days,  at  6  per  cent. 

J  of  18  days=6. 
300  4yr.  7mo.  18da.=:55.6mo. 

6 


$]  8.00=int.  for  1  year.     2)55.6,  int.  at  812  a  year. 
US=ll  times  $12.  278 

$83.4  Ans. 

If  the  interest  was  $12  a  year,  $55.<50  would  be 
the  answer ;  because  55.6  is  the  time  reduced  to 
months  ;  but  it  bears  $18  a  year,  or  1 J  times  12  j 
hence  1^  times  55.6  gives  the  interest  at  once. 

Example  2. — Kequired  the  interest  of  $150 
for  3  year^,  9  months,  and  27  days,  at  4  per  cent. 

J  of  27  days=9. 

150  3yr.  9mo.  27da.— 45.9mo. 

4  2)45.9,  int.  at  $12  a  year. 


$6.00=int.  for  1  year.     $22.95  Ans. 
$6=^  times  $12. 

If  the  interest  was  $12  a  year,  $45.90  would  be 
the  answer  j  because  245.9  is  the  time  reduced  to 
months;  but  it  bears  $6  a  year,  or  J  times  12,* 
hence  J-  times  45.9  gives  the  interest  at  once. 


MERCHANTS    METHOD  OF  COMPUTING  xNT.     Ill 

MERCHANTS'  METHOD 

OP 

COMPUTING    INTEREST. 

FOR  YEARS,  MONTHS,  AND  DATS. 

The  computation  of  simple  interest,  where  the 
time  consists  of  years,  months,  and  days,  is  quite 
difficult.  Taking  the  aliquot  parts  for  the  differ- 
ent portions  of  time  almost  invariably  involves  the 
calculator  in  fractions,  and,  unless  he  is  well  versed 
in  vulgar  fractions  he  will  not  be  able  to  arrive  at 
the  correct  result.  We  have  three  bases  by  which 
we  compute  interest  at  different  rates  per  cent, 
and  by  which  we  are  enabled  to  entirely  avoid  the 
use  of  fractions.  These  three  bases  are  each  obtained 
different  from  the  other,  and  consequently  we  have 
three  rules  for  computing  interest :  one  at  a  base  of 
one  per  cent.,  a  second  at  a  base  of  twelve  per 
sent.,  and  a  third  at  a  base  of  thirty-six  per  cent. 
KuLE  for  computing  interest  at  1  per  cent. : 
Take  one-tJurd  of  the  number  of  days  and  annex 
to  the  numher  of  months ;  divide  the  nnmher  thm 
formed  hy  \2\  annex  the  quotient  thus  obtained  to 
the  number  of  years^  and  multiply  tJie  principal  by 
this  number ;  if  the  principal  contains  cents,  point 
off  five  decimal  pluccs  ;  if  not^  point  off  three  deci- 


112  ORTONS'  LIGHTNING  CALCULATOR. 

mal  places;  this  will  give  the  interest  at  one  per 
cent.  For  any  other  rate  per  cent.^  multiply  the  in- 
terest at  one  per  cent,  hy  the  required  rate  per  cent. 

Remark. — This  rule  applies  to  all  problems  in 
interest  where  the  days  are  divisible  by  3,  and  this 
number,  annexed  to  the  number  of  months,  divisi- 
ble by  12. 

EXAMPLE. 

Required  the  interest  on  $112,  at  1  per  cent.,  for 
3  years,  3  months  and  18  days. 

SOLUTION. 

Take  one-third  of  the  number  of  days,  J  of  18 
=6,  annex  this  number  to  the  months  given,  36, 
divide  this  number  by  12,  36-i-12=3,  annex  this 
number  to  the  year  gives,  33,  multiply  the  princi- 
pal by  33,  $112X33=3.G9  6,  point  off  three  deci- 
mal places,  and  we  have  the  required  interest, 
$3.69  6. 

EXAMPLE. 

Required  the  interest  on  $125  12,  at  7  per  cent., 
for  2  years,  8  months  and  12  days. 

SOLUTION. 

Take  one-third  of  the  number  of  days,  J 
of  12=::4,  annex  this  number  to  the  number  of 
months  we  have   84,  divide  this  number   by  12, 


MEIICUANTS'  METUOD  OF  COMPUTING  INT.     113 

8-1 T- 12=7,  annex  this  number  to  the  $125  12 

number  of  years  we  have  27,  multiply  27 

the    principal    by   this    number,  and 

point  off  five  decimal  places,  and  you  3.37824 

have  the  interest  at  one  per  cent.;  mul-  7 

tiply  this  interest  by  7,  and  you  have 

the  interest  at  7  per  cent.,  the  required  $23  .6-1768 

rate. 

EXAMPLE. 

Required  the  interest  on  $1,023,  at  8  per  cent., 

for  1  year,  9  months  and  18  days. 

SOLUTION. 

Take  one-third  the  number  of  days  and  annex 

to   the  number  of  months,  J  of   18=6,  we  have 

96-^12=8,  annex  this  number  to  the  years       $1023 

we  have  18,  multiply  the  principal  by  18 

this  number,  and  point  of  three  decimal     

places,  which  gives  the  interest  at  1  per  $18  .414 
cent.;    multiply  the  interest  at  one  per  8 

cent,  by  8,  and  you  have  the  required  in 

teiest.  $147  .312 

Remarh. — This  rule  will  apply  to  all  problems 
in  interest  if  one-third  of  the  number  of  the  days 
be  taken  decimally  and  annexed  to  the  number  of 
months,  and  this  number,  divided  by  12,  carried 
out  decimally.  But  this  makes  the  multiplier 
very  large ;  hence,  to  avoid  this  large  number  in 


114         orton's  lightning  calculatcr. 

the  multiplier,  where  the  days  are  divisible  by  3, 
and  this  number,  annexed  to  the  months,  is  not 
divisible  by  12,  we  use  the  following  rule,  called 
our  base  at  12  per  cent. : 

lluLE. — Reduce  the  years  to  months^  add  in  the 
months,  taJce  one-third  of  the  number  of  days  and 
annex  to  this  number,  midtiply  the  principal  by  the 
number  thus  formed;  if  there  are  cents  in  the  prin- 
cipal, point  off  five  decimal  places  ;  if  there  are  no 
cents  in  the  principal,  point  off  three  decimal  places  ; 
this  gives  the  interest  at  12  per  cent.  For  any  other 
rate  per  cent.,  take  such  part  of  the  base  before  mul- 
tiplying as  the  required  rate  is  a  part  of  12. 

EXAMPLE. 
Required  the  interest  on  S123,  at  12  per  cent., 
for  2  years,  2  months  and  six  days. 
SOLUTION. 
Reduce   the   2   years   to   months   gives   us   24 
months,  add  on  the  2  months  gives  us  26 
months,  take  one-third  of  the  days,  J  of        $123 
6=2,  annexed   to   the  26  months   gives  262 

262,  which  constitutes  the  base ;  multiply     

the  principal  by  this  base,  and  you  have  $32  .226 
the  intcresi  at  12  per  cent. 

EXAMPLE. 
Required  the  interest  on  $144,  at  6  per  cent.,  for 
4  years,  5  months  and  12  days. 


MEECUANTS*  METHOD  OP  COMPUIINQ  INT.     115 

SOLUTION. 

Reduce  the  4  years  to  months  gives  48  months^ 
add  in  the  5  months  gives  53  months,  take  one- 
third  of  the  days  and  annex  to  the  number  of 
months,  J  of  12=4.  annex  to  the  53  months,  534 ; 
this  number  multiplied  into  the  principal  would 
give  the  interest  at  12  per  cent.  But  wc  want  it 
at  6  per  cent.  We  will  now  take  such  part  of 
either  principal  or  base  as  6  is  a  part  of  12 ;  6  is 
J  of  12,  therefore  we  will  take  ^  of  144=72 
one-half  of  the  principal,  and  mul-      «»  534 

tiply  it   by  the  base,  which  will 

give  the  interest  at  6  per  cent.  $38,448 

EXAMPLE. 

Required  the  interest  on  $347  25,  at  8  per  cent., 
for  2  years,  3  months  and  9  days. 
SOLUTION. 

Reduce  the  2  years  to  months,  24  months,  add 
the  3  months,  27  months,  take  one-third  of  the 
days,  J  of  9=3,  annex  to  the  months,  273,  the 
base;  this,  multiplied  into  the  principal,  would 
give  the  interest  at  12  per  cent.  But  we  want  the 
interest  at  8  per  cent ;  we  will  take 
two -thirds  of  the  base  before  multiply-  $347  25 
ing:    f   of    273=182;    the   principal  182 

multiplied  by  this  number  gives  the 

interest  at  8  per  cent.  $63.19950 

Remark. — This  base  is  used  where  the  days  are 
divisible  by  3,  and  the  number  formed  by  annex- 


116  ©ETON'S  LIGHTNING  CALCULATOR. 

ing  one-third  of  the  days  to  the  months  not  divisi- 
ble by  12.  We  now  come  to  time  in  which  neithei 
days  nor  months  are  divisible.  Where  such  time 
as  this  occurs,  we  uSe  a  base  at  36  per  cent. 

Rule. — Reduce  the  time  to  days^  hy  multiplying 
the  years  hy  12,  adding  in  the  months^  if  any,  and 
multiplying  this  number  hy  30,  adding  in  the  days, 
if  any;  multiply  the  principal  hy  this  number, 
pointing  off  5  decimal  places,  where  cents  are  given 
in  the  principal,  and  3  places  where  no  cents  are 
given.     This  will  give  the  interest  at  36  per  cent. 

EXAMPLE. 
Required  the  interest  on  $144,  at  36  per  cent., 
for  3  years,  2  months  and  2  days. 
SOLUTION. 
Reduce  the  time  to  days  gives   1142         $144 
days ;  multiply  the  principal  by  this  base,         1142 

and  you  have   the   interest   at   36    per 

cent  $164,448 

EXAMPLE. 

Required  the  interest  on  $144,  at  9  per  cent.,  lor 
5  years,  7  months  and  5  days. 
SOLUTION. 

Reduce  the  time  to  days  gives  2,015  days ;  if 
we  multiply  the  principal  by  this  base,  we  would 
get  the  interest  at  36  per  cent.;  but  we  want  it  at 
9   per  cent.     We  can  take  such  part  of   either 


merchants'  method  of  computing  int.  117 

principal  oi  6ase  as  9  is  a  part  of  36  before  multi- 
plying ;  9  is  J-  of  36 ;  we  will  take  J^  of  tlie  prin- 
cipal, it  being  divisible  by  4 ;  J  of  144=36,  2015 
which,  multiplied  into  the  base,  will  give  36 
the  interest  at  9  per  cent.,  by  pointing  off  

3  decimal  places.  $72,540 

EXAMPLE. 

Bequired  the  interest  on  $875  15,  at  6  per  cent., 
for  5  years,  7  months  and  12  days. 
SOLUTION. 

Reduce  the  time  to  days  gives  2022  days ;  6  h 
J  of  36 ;  take  one  sixth  of  the  base, 
J  of  2022=337;  multiply  the  prin-         $875  15 
cipal  by  this  number,  point  off"  5  dec-  337 

imal  places,  and  you  have  the  interest 

at  6  per  cent.,  the  required  rate.  $294.92555 

Remark. — We  have  now  fully  explained  our 
method  of  computing  interest  at  the  three  different 
bases.  Any  and  every  problem  in  interest  can  be 
solved  by  one  of  these  three  bases.  Some  prob- 
lems can  be  solved  easier  by  one  base  than  another. 
Where  the  days  are  divisible  by  3,  and  their  num- 
ber, annexed  to  the  months,  divisible  by  12,  it  is 
the  shortest  and  best  method  to  use  the  base  at  1 
per  cent.  By  using  one  or  the  other  of  these  three 
bases,  the  student  can  avoid  the  use  of  vulgar 
fractions.  The  student  must  study  these  three 
principles  carefully,  and  learn  to  adopt  readily  the 
base  best  suited  to  the  problem  to  be  solved. 


118         orton's  LianTiNQ  calculator. 


PARTIAL  PAYMENTS 


To  compute  interest  on  notes,  bonds,  and  mort- 
gages, on  whicli  partial  payments  have  been  made, 
two  or  three  rules  are  given.  The  following  is 
called  the  common  rule,  and  applies  to  cases  where 
the  time  is  short,  and  payments  made  within  a  year 
of  each  other.  This  rule  is  sanctioned  by  custom 
and  common  law;  it  is  true  to  the  principles  of 
simple  interest,  and  requires  no  special  enactment. 
The  other  rules  are  rules  of  law,  made  to  suit  such 
cases  as  require  (either  expressed  or  implied)  an- 
nual interest  to  be  paid,  and  of  course  apply  to  no 
business  transactions  closed  within  a  year. 

E-ULE. —  Compute  the  interest  of  the  ^principal  sum 
for  the  whole  time  to  the  day  of  settlement,  and  find 
the  amount.  Compute  the  interest  on  the  several  pay- 
ments, from  the  time  each  was  p<iid  to  the  day  of 
settlement;  add  the  several  payments  and  the  inter- 
est on  each  together,  and  call  the  sum  the  amount  of 
the  payments.  Subtract  the  amount  of  the  payments 
from  the  amount  of  tJie  principal,  will  leave  the  sum 
due. 


PARTIAL   PAYMENTS.  119 

EXA^IPLES. 
1.  A  gave  his  note  to  B  for  $10,000 ;  at  the  end 
of  4  months,  A  paid  $6,000;  and  at  the  expiration 
of  another  4  months,  he  paid  an  additional  sura  of 
$3,000 ;  how  much  did  he  owe  B  at  the  close  of  the 
year? 

By  the  Common  Rule, 

Principal $10,000 

Interest  for  the  whole  time 600 


Amount $10,600 

1st  payment $6,000 

Interest,  8  months      240 

2d  payment 3,000 

Interest,  4  months        60 


Amount $9,300  9,300 


Due $1300 

PROBLEMS   IN  INTEREST. 

There  are  four  parts  or  quantities  connected 
with  each  operation  in  interest:  these  are,  the 
Principal,  Rate  per  cent.y  Time^  Interest  or  Amount. 

If  any  three  of  tLcm  are  given  the  other  may  be 
found. 

Principal,  interest,  and  time  given,  to  find  the 
rate  per  cent. 

1.  At  what  rate  per  cent,  must  $500  be  put  on 
interest  to  gain  $120  in  4  years? 


120       orton's  lightning  calculator. 


Operation. 

By  analysis. 

$500 

The   interest  of 

.01 

$1    for    the    given 

time  at  1  per  cent. 

5.00 

is   4   cents.     $500 

4 

will    be  500   times 

asmiich=500X.04 

20.00)120 

.00(6  per  cent.. 

,Ans. 

=$20.00.    Then  if 

120.00 

$20  give  1  percent., 

$120  will  give  \%o 

=6  per  cent. 

RnLE. — Dimde  the  given  interest  hy  the  interest 

of  the  given  sum  at  1  per  cent,  for  the  given  time.j 

and  the  quotient  will  he  the  rate  j)er  cent,  required 

Principal,  interest,  and  rate  per  cent,  given,  to 

find  the  time. 

2.  How  long  must  $500  be  on  interest  at  6  per 
cent,  to  gain  $120  ? 

Operation  By  analysis. 

$500  We  find  the  in- 

.06  terest  of  $1.00    at 

the  given  rate   for 

30.00)120.00(4  years,  J.n5.  1  year  is  6  cents 
120.00  $500,  will  therefore 
be    500    times    aa 


much=500X  .06=$30.00.  Now,  if  it  take  1  year 
to  gain  $30,  it  will  require  V*o°  to  gain  $120=4 
years,  Ans, 


PARTIAL    PAYMENTS.  121 

Rule. — Divide  the  given  interest  hy  tJie  interest 
of  the  princijpal  for  1  year^  and  the  quotient  is  the 
time. 

Given  the  amount,  time,  end  rate  per  cent.,  to 
find  the  principal. 

Rule. — Divide  the  given  amount  hy  the  amount 
o/^l,  at  the  given  rate  per  cent.^  for  the  given  time. 

Remark. — This  rule  is  deduced  from  the  fact 
that  the  amount  of  diflferent  principals  for  the  same 
time  and  at  the  same  rate  per  cent.,  arc  to  each 
other  as  those  principals. 

BANK    DISCOUNT. 

Banh  Discount  is  the  sum  paid  to  a  hank  for  the 
payment  of  a  note  before  it  becomes  due. 

The  amount  named  in  a  note  is  called  the  face 
of  the  note.  The  discount  is  the  interest  on  the 
face  of  the  note  for  3  days  more  than  the  time 
specified,  and  is  paid  in  advance.  These  3  days 
are  called  days  of  grace,  as  the  borrower  is  not 
obliged  to  make  payment  until  their  expiration. 
Hence,  to  compute  bank  discount,  we  have  the  fol- 
lowing 

Rule. — Find  the  interest  on  the  face  of  the  note 
for  3  days  more  than  the  time  specif  ed ;  this  will 
he  the  discount.  From  the  face  of  the  note  deduct 
the  discount,  and  tht  remainder  will  he  the  present 
VALUE  of  the  note. 


122       orton's  lightning  calculator. 

disco  itnt,  or  counting  back. 
The  object  of  discount  is  to  show  us  what  al- 
lowance should  be  made  when  any  sum  of  money 
is  paid  before  it  becomes  due. 

The  present  worth  of  any  sum  is  the  principal 
that  must  be  put  at  interest  to  amount  to  that  sum 
in  the  given  time.  That  is,  SI 00  is  the  present  worth 
of  $10G  due  one  year  hence;  because  $100  at  6  per 
cent,  will  amount  to  $106  j  and  S6  is  the  discount. 
1.  What  is  the  present  worth  of  $12.72  due  one 
year  hence  ? 

First  method.  Second  method. 

$12.72  $ 

100  1.06)12.72($12  Ans, 

10.6 

106)1272.00($12  Am.  

106  2.12 

2.12 


212 
212 


As  $100  will  amount  to  $106  in  one  year  at  6 
per  cent.,  it  is  evident  that  if  ig§  of  any  sum  be 
taken,  it  will  be  its  present  worth  for  one  year,  and 
that  y^g  will  be  the  discount.  And  as  $1  is  the 
present  worth  of  $1.06  due  one  year  hence,  it  is 
evident  that  the  present  worth  of  $12.72  must  be 
equal  to  the  number  of  times  $12.72  will  contain 
$1.06. 


EQUATION   OF   PAYMENTS.  123 

Rui-E. — Divide  the  given  sum  hy  the  amount  of 
$1  for  the  given  rate  and  time,  and  the  quotient  will 
he  the  'present  worth.  Jf  the  present  worth  he  suh- 
tracted  from  the  given  sum,  the  remainder  will  he  the 
discount. 


EQUATION  OF  PAYMENTS. 

Equation  op  Payments  is  the  process  of  find- 
ing the  equalized  or  average  time  for  the  payment 
of  several  sums  due  at  different  times,  without  loss 
to  either  party. 

To  find  the  average  or  mean  time  of  payment, 
when  the  several  sums  have  the  same  date. 

Rule. — Multiply  each  payment  hy  the  time  that 
must  elapse  hefore  it  hecomes  due;  then  divide  the 
sum  of  these  products  hy  the  sum,  of  the  payments, 
and  the  quotient  will  he  the  averaged  time  required. 

Note. — When  a  payment  is  to  be  made  down,  it 
has  no  product,  but  it  must  be  added  with  the 
other  payments  in  finding  the  average  time. 

Example  1. — I  purchased  goods  to  the  amount 
of  $1200;  $300  of  which  I  am  to  pay  in  4  months, 
$tOO  in  5  months,  and  $500  in  8  months.  IIow 
long  a  credit  ought  I  to  receive,  if  I  pay  the 
whole  fiUm  at  once?  Ans.  G  mouths. 


124       orton's  lightning  calculator. 

Mo.  Mo,  r    A  credit  on  $300  for  t  inonthB  ii 

4.  -OAA 1  OAA  ■<  the  samo  as  the  credit  on  Jl   fo« 

y^oy)[):=i.^\JO  h200month8. 

r    A  credit  on  $400  for  5  months  it 
5  "< 400=2000  ^thesarao  as  the  credit  on  $1  for 

^2000  months. 

Qv^  rnn 4(\f\(\  \     A  credit  on  $500  for  8  months  is 

°AJ^^ — -iUUU  J  the  same  as  the  credit  on  $1  for 

.      (_4000  months. 

1  ctnfw  ^70AA  /£»  Therefore,   I    should    have    the 

IZUUJ  <ZUU  (O  mo.       name  credit  as  a  credit  on  $1  for 

79rtn  '^^^^    months,   and  on   $1200,  the 

t  ^^yf  whole  sum,  one-twelfth  hundredth 

.  part  of  7200  months,   which  is  6 

months. 

This  rule  is  the  one  usually  adopted  by  mer- 
chants, although  not  strictly  correct,  still,  it  is  suf- 
ficiently accurate  for  all  practical  purposes.. 

To  find  the  average  or  mean  time  of  payme-nt, 
when  the  several  sums  have  different  dates. 

Example  1. — Purchased  of  James  Brown,  at 
pundry  times,  and  on  various  terms  of  credit,  as  by 
the  statement  annexed.  When  is  the  medium  time 
of  payment? 

Jan.      1,  a  bill  am'ting  to  $360,  on  3  months'  credit. 
Jan.    15,  do.       do.  186,  on  4  months' credit. 

March  1,  do.       do.  450,  on  4  months' credit. 

May    15,  do.       do.  300,  on  3  months' credit 

June  20,  do.       do.  500,  on  5  months'  credit. 

Ans.  July  24th,  or  in  115  da. 
Duo  April    1,  $360 

May   15,    186X  44=    8184 

July     1,    450X   91=  40950 

Aug.  15,    300X136=  40800 

Nov.  20,    500X233=116500 

1796V  into)2064;M(114|J«  dayi*. 


EQUATION   OP   PAYMENTS.  125 

We  first  find  the  time  wlien  eacli  of  tlie  bill3 
will  become  due.  Then,  since  it  will  shorten  the 
operation  and  bring  the  same  result,  we  take  the 
time  when  the  first  hill  becomes  due^  instead  of  its 
datc^  for  the  period  from  which  to  compute  the 
average  time.  Now,  since  April  1  is  the  period 
from  which  the  average  time  is  computed,  no  time 
will  be  reckoned  on  the  first  bill,  but  the  time  for 
the  payment  of  the  second  bill  extends  44  days  be- 
yond April  1,  and  we  multiply  it  by  44. 

Proceeding  in  the  same  manner  with  the  remain- 
ing bills,  we  find  the  average  time  of  payment  to 

114  days  and  a  fraction,  from  April  1,  or  on  the 
24th  of  July. 

Rule. — Find  the  time  when  each  of  the  sums  he- 
comes  due,  and  multij^Jy  each  sum  hy  the  numher  of 
days  from  the  time  of  the  earliest  imyment  to  the 
payment  of  each  sum  respectively.  Then  proceed  as 
in  the  last  rule,  and  the  quotient  will  he  the  aver- 
age time  required^  in  days,  from  the  earliest  pay- 
ment. 

Note. — Nearly  the  same  result  may  be  obtained 
by  reckoning  the  time  in  months. 

In  mercantile  transactions  it  is  customary  to  give 
a  credit  of  from  3  to  9  months,  on  bills  of  sale. 
Merchants  in  settling  such  accounts,  as  consist  of 
various  items  of  debit  and  credit  for  different  times, 
generally  employ  the  following : 


126  ORTON'S  LIGHTNINQ  CALCULATOR. 

Rule. — Place  on  the  dehtor  or  credit  side,  such  a 
sum,  (which  may  he  called  merchandise  BALANCE,) 
a&  will  halance  the  account. 

Multiply  the  number  of  dollars  in  each  entry  hy 
the  number  of  days  from  the  time  the  entr-y  was  made 
to  the  time  of  settlement;  and  the  Merchandise  bal- 
ance by  the  number  of  days  for  which  credit  was 
given.  Then  multiply  the  difference  between  the  sum 
of  the  debitf  and  the  sum  of  the  credit  products,  by 
the  interest  0/  $1  for  1  day;  this  product  will  be 

the  INTEREST  BALANCE. 

When  the  sum  of  the  debit  products  exceed  the  sum 
of  the  credit  products,  the  interest  balance  is  in  favor 
of  the  debit  side ;  but  when  the  sum  of  th^  credit 
products  exceed  the  sum  of  the  debit  products,  it  is  in 
favor  of  the  credit  side.  Now  to  the  merchandise 
balance  add  the  interest  balance,  or  subtract  it^  as  the 
case  may  require,  and  you  obtain  tJie  CASH  BAL- 
ANCE. 

A  has  with  B  the  following  account : 

1849.  Dr.    I      1849.  Or. 


Jan.  2.     To  merchandise,    8200      Feb.  20.  By  merchandise,  8100 
April  20.  "  "  400 1    May.  10.  "  ♦•  300 

If  interest  is  estimated  at  7  per  cent.,  and  a 
credit  of  60  days  is  allowed  on  the  different  sums, 
what  is  the  cash  balance  August  20,  1849  ? 

Am.  206.54. 

EXPL4NATI0N. — "Without  interest  the  cash  bal- 
ance would  be  $200. 


EQUATION    OF   PAYMENTS.  127 

If  no  credit  had  been  given,  the  debits  sliould 
be  increased  by  tbe  interest  of  $200  for  230  days, 
at  7  percent.;  and  the  interest  of  $400  for  122 
days,  at  7  per  cent.  The  credits  should  be  increas- 
ed by  the  interest  of  $100  for  181  daj^s,  at  7  per 
cent.,  and  the  interest  of  $300  for  102  days,  at  7 
per  cent. 

Since  a  credit  of  60  days  is  given  on  all  sums,  it 
is  evident  by  the  above  calculation,  that  we  should 
increase  the  debits  by  the  interest  of  the  sum  of 
the  debits,  $600,  for  60  days  more  than  justice  re- 
quires. Also,  that  we  should  increase  the  credits 
by  the  interest  of  the  sum  of  the  credits,  $400,  for 
60  days  more  than  we  should  do. 

Now,  instead  of  deducting  these  items  of  inter- 
est from  the  amount  of  debit  and  credit  interests, 
it  is  plain  that  it  will  be  more  convenient  and 
equally  just,  to  diminish  the  debit  interest  of  the 
merchandise  halance  for  60  days,  whidi  can  be 
most  readily  accomplished  by  adding  the  interest 
on  the  merchandise  balance  for  60  days,  to  the 
credit  items  of  interest. 

From  which  we  discover  that  the  interest  halance 
is  equal  to  the  diilerence  between  the  sum  of  the 
debit  interests,  and  the  sum  of  the  credit  interests 
increased  by  the  interest  of  the  merchandise  bal- 
ance for  the  time  for  which  credit  was  given. 


128        orton's  lightning  calculator. 
Operation. 

DEBITS.  CREDITS. 

Z        Days.  S        Days. 

200X  230=46000  J  OOx  181=18100 

400X122=48800  300x102=30600 

Balance,  200X  60=12000 


94800 

60700  60700 


0.07 

X 34100=$6.54  Interest  balance,  yearly, 

365 
Therefore,  the  foregoing  account  becomes  bal- 
anced as  follows : 


1849.  Dr. 

/an .     2.  To  Merchandise,  $200.00 
April 20.  '«  "  400.00 

Aug.  20.  **  balance  of  int.       6.54 


{606.54 


1849  Or. 

Fob.  20.  By  Merchandise,  $100.00 
May.  10.  "  "  300.00 

Aug.  20.  «•  balance,  206.54 

J606.64 


Aag.  20.  '<  Cash  balance,  $206.54 

Note. — It  is  customary  in  practice,  when  the 
number  of  cents  in  any  of  the  entries,  are  less  than 
50,  tc  omit  them,  and  to  add  $1  when  they  are  50, 
or  more. 


SQUARE  AND  CUBE  ROOTS.        1^9 

SQUARE  AND  CUBE  ROOTS. 

To  work  the  square  and  cube  roots  with  ease  and 
lacility,  the  pupil  must  be  familiar  with  the  follow- 
ing properties  of  numbers : 

Their  importance  can  not  be  exaggerated  if  we 
wish  to  insure  skill  or  even  sound  information  on 
this  subject. 

I.  A  square  number,  multiplied  by  a  square 
number,  the  product  will  be  a  square  number. 

II.  A  square  number,  divided  by  a  square  num- 
ber, the  quotient  is  a  square. 

III.  A  cube  number,  multiplied  by  a  cube,  the 
product  is  a  cube. 

IV.  A  cube  number,  divided  by  a  cube,  the  quo- 
tient will  be  a  cube. 

V.  If  the  square  root  of  a  number  is  a  compos- 
ite number,  the  square  itself  may  be  divided  into 
integer  square  factors ;  but  if  the  root  is  a  prime 
numherj  the  square  can  not  be  separated  into  square 
factors  witlwut  fractions. 

VI.  If  the  unit  figure  of  a  square  number  is  5,  we 
may  multiply  by  the  square  number  4,  and  we  shall 
have  another  square,  whose  unit  period  will  be 
ciphers. 

VII.  If  the  unit  figure  of  a  cube  is  5,  we  may 
Multiply  by  the  cube  number  8,  and  produce  an- 
other  cube,  whose  unit  period  will  be  ciphers. 


130        orton's  lightning  calculator. 

K.  B.  If  a  supposed  cube,  whose  unit  figure  is 
5,  be  multiplied  by  8,  and  tbe  product  does  not  give 
three  ciphers  on  the  right,  the  number  is  not  a  cube. 

We  present  the  following  table,  for  the  pupil  to 
compare  the  natural  numbers  with  the  unit  figure 
of  their  squares  and  cuhes,  that  he  may  be  able  to 
extract  roots  hy  inspection. 


Numbers 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

Squares 

1 

4 

9 

16 

25 

36 

49 

64 

81 

100 

Cubes 

1 

8 

27 

64 

125 

216 

343 

512 

729 

1000  1 

EXERCISES  FOR  PRACTICE. 

1.  What  is  the  square  root  of  625?    Ans.  25 
If  the  root  is  an  integer  number^  we  may  know, 

by  the  inspection  of  the  table,  that  it  must  be  25, 
as  the  greatest  square  in  6  is  2,  and  5  is  the  only 
6gure  whose  square  is  5  in  its  unit  place. 

Again,  take  625 

Multiply  by  4     4  being  a  square. 

2500 
The  square  root  of  this  product  is  obviously  50; 
but  this  must  be  divided  by  2,  the  square  root  of 
^,  which  gives  25,  the  root. 

2.  What  is  the  square  root  of  6561  ?     Ans.  81. 
As  the  unit  figure^  in  this  example,  is  1,  and  in 


SQUARE  AND  CUBE  ROOTS.        131 

tlie  Hue  of  squares  in  the  table,  we  find  1  only  at  1 
and  81,  we  will,  therefore,  divide  6561  by  81,  and  we 
find  the  quotient  81  j  81  is,  therefore,  the  square  root. 

3.  What  is  the  square  root  of  106729?  Ans.  327. 
As  the  unit  figure,  in  this  example,  is  0,  if  the 

number  is  a  square,  it  must  divide  by  either  9,  or 
49.  After  dividing  by  9  we  have  11881  for  the 
other  factor,  a  prime  number,  therefore  its  root  is  a 
prime  number=:109.  109,  multiplied  by  3,  the 
root  of  9,  gives  327  for  the  answer. 

4.  AVhat  is  the  root  of  451584?         Ans.  672. 
As  the  unit  figure  is  4,  and  in  the  line  of  squarea 

we  find  4  only  at  4  and  64,  the  above  number,  if  a 
square^  must  divide  by  4,  or  64,  or  by  both. 

We  will  divide  it  by  4,  and  we  have  the  factors 
4  and  112896.  This  last  factor  closes  in  6  ;  there- 
fore, by  looking  at  the  table,  we  see  it  must  divide 
by  1Q,  or  36,  etc. 

We  divide  by  36,  and  we  have  the  factors  36  and 
3136;  divide  this  last  by  16,  and  we  have  16  and 
196 ;  divide  this  last  fraction  by  4,  and  we  have  4 
and  49. 

Take  now  our  divisors,  and  last  factor,  49,  and 
we  have  for  the  original  number  the  product  of 
4X36X16X4X49;  the  roots  of  which  are  2x6 
X 4X2X7,  the  products  of  which  are  672,  the 
answer 

5.  Extract  the  square  root  of  2025.     Ans.  45. 


132       okton's  lightning  calculator. 

1st.  Divide  by  the  square  number  25,  and  wd 
find  the  two  factors,  25x81,  as  equivalent  to  the 
given  number.  Roots  of  these  factors,  5x9^^45, 
the  answer. 

Again,  multiply  by  the  square  number  4,  when 
a  number  ends  in  25,  and  we  have  8100,  root  90, 
half  of  which,  because  we  multiplied  by  4,  the 
square  of  2,  is  45,  the  answer. 

Prohlems  on  the  Right-angled  Triangle. 

1.  The  top  of  a  castle  is  45  yards  high,  and  is 
surrounded  with  a  ditch,  60  yards  wide ;  required 
the  length  of  a  ladder  that  will  reach  from  the 
outside  of  the  ditch  to  the  top  of  the  castle. 

Ans.  75  yards. 

This  is  almost  invariably  done  by  squaring  45 
and  60,  adding  them  together,  and  extracting  the 
square  root ;  but  so  much  labor  is  never  necessary 
when  tlie  numbers  have  a  common  divisor^  or  when 
the  side  sought  is  expressed  by  a  composite  number. 

Take  45  and  60 ;  both  may  be  divided  15,  and 
they  will  be  reduced  to  3  and  4.     Square  these, 
9+16=25.     The  square  root  of  25  is  5,  which, 
multiplied  by  15,  gives  75,  the  answer. 
Abbreviations  in  Cube  Root 

1.  Wha^  is  the  cube  root  of  91125  ?     Am.  45. 
Multiply  by  8 

729000 


SQUARE  AND  CUBE  ROOTS.        133 

New,  729  being  the  cube  of  9,  the  root  of 
729000  is  90 ;  divide  this  by  2,  the  cube  root  of  8, 
and  we  have  45,  the  answer. 

When  it  is  requisite  to  multiply  several  numbers 
together  and  extract  the  cube  root  of  their  pro- 
duct, try  to  change  them  into  cuhe  factors  and  ex- 
tract the  root  be/ore  multiplication. 

EXAMPLES. 

1.  What  is  the  side  of  a  cubical  mound  equal 
to  one  288  feet  long,  216  feet  broad,  and  48  feet 
high? 

The  common  way  of  doing  this,  is  to  multiply 
these  numbers  together  and  extract  the  root — a 
lengthy  operation.  Bat  observe  that  216  is  a  cube 
number,  and  288=2x12X12,  and  48=4X12; 
therefore  the  whole  product  is  216x8Xl2Xl2X 
12.  Now,  the  cube  root  of  216  is  6,  of  8  is  2,  and 
of  12'  is  12,  and  the  product  of  6X2X12=144, 
the  answer. 

2.  Required  the  cube  root  of  the  product  of 
448X392  the  short  way.  Ans.  56. 

We  can  extract  the  root  of  cuhe  numhers  hy  in- 
spcction  when  they  do  not  contain  more  than  two 
periods. 


134       okton's  lightning  calculator. 

RuLK. — As  there  will  be  two  figures  in  the  root,  the  first 
may  easily  be  found  mentally,  or  by  the  Table  of  Powers ',  and 
if  tf^  unit  figure  of  the  power  be  1,  the  unit  figure  in  the  root 
will  be  1 ;  and  if  it  be  8,  the  root  will  be  tioo ;  and  if  7,  it 
will  be  3 ;  and  if  the  unit  of  the  power  be  6,  the  unit  of  the 
root  will  be  6 ;  and  if  5,  it  will  be  6]  if  3,  it  will  bel ;  (/"  2, 
it  vnll  be  8 ;  and  if  the  unit  of  the  power  be  9,  the  unit  of  the 
root  will  be  9.  This  will  appear  evident  by  inspecting  the 
Table  of  Powers. 

EXAMPLES. 

Find  the  cube  root  of  195112.  This  number 
consists  of  two  periods.  Compare  the  superior 
period  with  the  cubes  in  the  table,  and  we  find  that 
195  lies  between  125  and  216.  The  cube  root  of 
the  tens,  then,  must  be  5.  The  unit  figure  of  the 
given  cube  is  2 ;  and  no  cube  in  the  table  has  2 
for  its  unit  figure,  except  512,  whose  root  is  8 ; 
therefore  58  is  the  root  required. 

What  is  the  cube  root  of  97336  ?        Ans  46. 

Explanation. — By  examining  the  left  hand 
period,  we  find  the  root  of  97  is  4,  and  the  cube  of 
4  is  64.  The  root  can  not  be  5,  because  the  cube 
of  5  16  125.  The  unit  figure  of  the  given  cube  is 
6 ;  and  no  cube  in  the  table  has  6  for  its  unit  fig- 
ure, except  216,  whose  root  is  6  ;  the  answer,  there- 
fore, is  46. 

The  number  912673  is  a  cube  ;  what  is  its  root? 

Ans.  97. 

Observe,  the  root  of  the  superior  period   must 


SQUARE   AND    CUBE   ROOTS.  135 

be  9,  and  the  root  of  the  unit  period  must  be  some 
number  which  will  give  3  for  its  unit  figure  when 
cubed ;  and  7  is  the  only  figure  that  will  answer. 

The  following  numbers  are  cubes ;  required  their 
roots. 

1.  What  is  the  cube  root  of  59319?      Ans.  39 

2.  What  is  the  cube  root  of  79507?      Ans.  43. 
a  What  is  the  cube  root  of  117649?    Ans.  49. 

4.  What  is  the  cube  root  of  110592?    Ans.  48. 

5.  What  is  the  cube  root  of  357911?    Ans.  71. 

6.  What  is  the  cube  root  of  389017  ?    Ans.  73. 

7.  What  is  the  cube  root  of  571787  ?    Ans.  83. 
When  a  cube  has  more  than  two  periods,  it  can 

generally  be  reduced  to  two  by  dividing  by  some 
one  or  more  of  the  cube  numbers,  unless  the  root 
is  a  prime  number. 

The  number  4741632  is  a  cube;  required  its 
root.  Here  we  observe  that  the  unit  figure  is  2 ; 
the  unit  figure  of  the  root  must  therefore  be  the 
root  of  512,  as  that  is  the  only  cube  of  the  9  dig- 
its whose  unit  figure  is  2.  The  cube  root  of  512 
is  8 ;  therefore  8  is  the  unit  figure  in  the  root,  and 
the  root  is  an  even  number,  and  can  be  divided  by 
2 ;  and  of  course  the  cube  itself  can  be  divided  by 
8,  the  cube  of  2.  8)4741632 


592704 
Now,  as  the  first  numbor  was  a  cube,  and  being 


136       orton's  lightning  calculator. 

divided  by  a  cube,  tlie  number  592704  must  be  a 
cube,  and  by  inspection,  as  previously  explained, 
its  root  must  be  84,  which,  multiplied  by  2,  gives 
168,  the  root  required. 

The  number  13312053  is  a  cube;  what  is  its 
root?  Ans.  237. 

As  there  are  three  periods,  there  must  be  three 
figures,  units,  tens,  and  hundreds,  in  the  root;  the 
hundreds  must  be  2,  the  units  must  be  7.  Let  ua 
then  divide  the  2d  figure,  or  the  tens,  in  the  usual 
wai/j  and  we  have  237  for  the  root. 

Again,  divide  13312053  by  27,  and  we  have 
493039  for  another  factor.  The  root  of  this  last 
number  must  be  79,  which,  multiplied  by  3,  the 
cube  root  of  27,  gives  237,  as  before. 

The  number  18609625  is  a  cube;  what  is  its 
root? 

As  this  cube  ends  with  5,  we  will  multiply  it 
by  8: 

18609625 
8 


148877000 
As  the  first  is  a  cube,  this  product  must  be  a  cube; 
and  as  far  as  labor  is  concerned,  it  is  the  same  aa 
reduced  to  two  periods,  and  the  root,  we  perceive 
at  once,  must  be  530,  which,  divided  by  2,  gives 
265  for  the  root  required. 


SQUARE  AND  CUBE  ROOTS.        137 

JS.  B. — If  a  number,  wliosc  unit  figure  is  5,  be 
multiplied  by  8,  and  docs  not  result  in  three  ciphers 
on  the  right,  the  number  is  not  a  cube. 

To  find  the  Approximate  Cube  Root  oj  Surds. 

KuLE. —  Tahe  the  nearest  rational  cithe  to  the  given 
number  J  and  call  it  the  assumed  cube;  or  assume  a 
root  to  the  given  number  and  cube  it.  Double  the 
assumed  cube  and  add  the  number  to  it;  also  double 
the  number  and  add  the  assumed  cube  to  it,  Tahe 
the  difference  of  these  sums^  then  say^  As  double  of 
the  assumed  cube^  added  to  the  number^  is  to  this  dif- 
ferencCy  so  is  the  assumed  root  to  a  correction. 

This  correction,  added  to  or  subtracted  from  the 
assumed  root,  as  the  case  may  require^  will  give  the 
cube  root  very  nearly. 

By  repeating  the  operation  with  the  root  last 
found  as  an  assumed  root,  we  may  obtain  results 
to  any  degree  of  exactness ;  one  operation,  how- 
ever, ig  generally  sufficient. 

EXAMPLES. 

1.  Bequired  the  cube  root  of  QQ. 

The  cube  root  of  64  is  4.  Now  it  is  manifest 
that  the  cube  root  of  66  is  a  little  more  than  4, 
and  by  taking  a  similar  proportion  to  the  preced- 
ing, we  have 

64X2=128     2X66=132 
QQ  64 

194  196:  :4:  to  root  of  66. 


138  ORTON's    .LIGHTNINa    CALCULATOR. 

Or,  194  :  2  :  :  4  :  to  a  correction 

194)8.0000(0.04124 
7  76 


240 
194 


460 
388 


720 
Therefore  the  cube  root  of  G6  is  4.04124. 
2.  Required  the  cube  root  of  123. 
Suppose  it  5  ;  cube  it,  and  we  have  125. 
Now  we  perceive  that  the  cube  of  5  being  greater 
than  123,  the  correction  for  5  must  be  sultracted. 

2X125=250     246 
Add  123     125 

As  373  :  371  :  :  5  :  root  of  123. 

Or,    373     :    2    :  :     5     :    correction  for  5 


373)10.0000(0.0268] 
7  46 


2  540  From  5.00000 

2  238  Take  0.02681 


3020         Ans.    4.97319 
2984 

3G0. 


SQUARE   AND   CUBE   ROOTS.  139 

3.  What  is  the  cube  root  of  28  ?    Ans.  3,03658+ 

4.  What  is  the  cube  root  of  26  ?    Ans.  2,96249+ 

5.  What  is  the  cube  root  of  214  ?  Ans.  5,98142  f 

6.  What  is  the  cube  root  of  346  ?  Ans,  9,02034+ 
The  above  being  very  near  integral  cubes—  that 

is,  28  and  26  are  both  near  the  cube  number  27, 
214  is  near  216,  etc.  All  numbers  very  near  cube 
numbers  are  easr/  of  solution. 

We  now  give  other  examples,  more  distant  from 
integral  cubes,  to  show  that  the  labor  must  be  more 
lengthy  and  tedious,  though  the  operation  is  the 
same. 

1.  What  is  the  cube  root  of  3214?  ^ns.  14,75758. 

Suppose  the  root  is  15 — its  cube  is  3375,  which, 
being  greater  than  3214,  shows  that  15  is  too  great  j 
the  correction  will  therefore  be  sub  tractive. 

By  the  rule,  9964  :  161  :  :  15.  0,243,  the 
correction. 

Assumed  root 15,0000 

Less 2423 

Root  nearly 14,7577 

Now  assume  14,7  for  the  root,  and  go  over  the 
operation  again,  and  you  will  have  the  true  root  to 
8  or  10  places  of  decimals. 

N.  B. — Boots  of  component  powers  may  be  ob- 
tained more  readily  thus : 

For  the  4th  root,  take  the  square  root  of  the 
square  root, 
w* 


140       orton's  lightning  calculator. 
APPLICATION  OF  THE  CUBE  EOOT. 

PRINCIPLES  ASSUMED. 

Spheres  are  to  each  other  as  the  cubes  of  their 
diameter. 

Cubes,  and  all  solids  whose  corresponding  parts 
are  similar  and  proportional  to  each  other,  are  to 
each  other  as  the  cubes  of  their  diameters,  or  of 
their  homologous  sides. 

1.  If  a  ball,  3  inches  in  diameter,  weigh  4  pounds, 
what  will  be  the  weight  of  a  ball  that  is  6  inches 
in  diameter?  Am.  321bs. 

2.  If  a  globe  of  gold,  1  inch  in  diameter,  be 
worth  $120,  what  is  the  value  of  a  globe,  3J  inches 
in  diameter?  -4rw.  $5145. 

Questions  Solved  by  the  Jtule  of  Three^ 
JDirect  or  Inverse, 

There  is  a  cistern  which  has  a  stream  of  water 
running  into  it;  it  has  10  cocks;  all  running  to- 
gether will  empty  it  in  2 J  hours ;  6  will  enjpty  it 
in  5 J  hours ;  how  long  will  it  take  3  to  empty  it  ? 

Ans.  55  hours. 

Note. — The  6  cocks  will  discharge  in  4J  hours 
what  the  10  cocks  will  in  2J  hours ;  therefore  it 
would  take  the  6  cocks  1 J  to  discharge  what  would 
run  into  the  cistern  in  3  hours :  therefore  it  would 
take  the  6  cocks  1111  to  discharge  what  would  run 


SQUARE  AND  CUBE  ROOTS.        141 

in  2J  hours ;  consequently,  2f  cocks  to  discliarge 
the  water  as  fast  as  it  run  in. 

There  is  a  stick  of  timber,  12  feet  long,  to  be 
carried  by  3  men :  one  carries  at  the  end,  the  other 
two  carry  by  a  lever ;  how  far  must  the  lever  be 
placed  from  the  other  end  that  each  may  carry 
equally  ?  Ans.  3  feet  from  the  end. 

Note. — All  bodies  gravitate  in  an  inverse  pro- 
portion to  the  distance  of  the  center  of  gravity. 

As  1  is  to  6,  the  center,  so  is  2  to  the  answer 
required. 

Man.  Feet.        Men. 

As  1     :    6    :  :    2 
1 

2)6 

3  Ant. 

A  stick  of  timber,  30  feet  long,  to  be  carried  by 
5  men :  two  carry  at  one  end,  the  other  three  by  a 
lever ;  how  far  from  the  center  must  the  lever  be 
placed  that  all  may  carry  equally  ? 

Men.       Feet.        Men. 
As  2    :     15     :  :    3 
2 

3)30 

10  Ans. 


142        orton's  lightning  calculator. 
MENSURATION  OR  PRACTICAL  GEOMETRY, 

MEASUREMENT   OF   GRINDSTONES. 

Grindstones  are  sold  by  the  stone,  and  their  con- 
tents found  as  follows:* 

KuLE. — To  the  whole  diameter  add  half  of  the 
diameter,  and  multiply  the  sum  of  these  by  the 
same  half  and  this  product  by  the  thickness ;  di- 
vide this  last  number  by  1728,  and  the  quotient  is 
the  contents  J  or  answer  required. 

EXAMPLES. 
What  are   the  contents  of  a  grindstone  24 
inches  diameter,  and  4  inches  thick 


24+12X24x4 
=1  stone.  Ans, 

1728 
2.  What  are  the  contents  of  a  grindstone  36 
inches  diameter,  and  4  inches  thick.    Ans.  2J  stone. 

Mensuration  of  Superficies  and  Solids. 
Superficial  measure  is  that  which  relates  to  length 
and  breadth  only,  not  regarding  thickness.  It  is 
made  up  of  squares,  either  greater  or  less,  accord- 
ing to  the  different  measures  by  which  the  dimen- 
sions of  the  figure  are  taken  or  measured.  Land 
is  measured  by  this  measure,  its  dimensions  being 

*24  inclies  in  diameter,  and  4  inches  thick  make  a  stone. 


MENSURATION  OR  PRACTICAL  GEOMETRY.    143 

usually  taken  in  acres,  rods,  and  links.  The  con- 
tents of  boards,  also,  are  found  by  this  measure, 
their  dimensions  being  taken  in  feet  and  inches. 
Because  12  inches  in  length  make  1  foot  of  long 
measure,  therefore  12X12=144,  the  square  inches 
in  a  superficial  foot,  etc. 

Note. — Superficial  means  lying  on  the  surface. 

To  find  the  area  of  a  square  having  equal  sides. 

Rule. — Multiply  the  side  of  the  square  into 
itself  and  the  product  will  be  the  area,  or  superfi- 
cial content  of  the  same  name  with  the  denoinina- 
Hon  taken,  whether  inches,  feet,  yards,  rods,  and 
links,  or  acres, 

EXAMPLES. 

1.  IIow  many  square  feet  of  boards  are  contain- 
ed in  the  floor  of  a  room  which  is  20  feet  square  ? 

20X20=400  feet,  the  answer. 

2.  Suppose  a  square  lot  of  land  measures  36 
rods  on  each  side,  how  many  acres  does  it  contain  ? 

30X36  =  1296  square  rods.     And 
1296-v-160=8  acres,  16  rods,  Ans. 

As  160  square  rods  make  an  acre,  therefore  we  di- 
vide 1296  by  IGO  to  reduce  rods  to  acres. 

N.  B. — The  shortest  way  to  work  this  example 
is,  to  cancel  36X36  with  the  divisor  160.  Arrange 
the  example  as  below ;  (divide  both  terms  by  4x4 :) 

36X36  9X9 

" same  as  ~ =8.1  acres,  or  Sac.  16  rods. 

160  10 


141        ohton's  lightning  calculator. 

To  measure  a  parallelogram  or  long  square, 

KuLE. — Multiply  the  length  by  the  breadth^  and 

the  product  will  be  the  area^  or  superficial  content^, 

in  the  same  name  as  that  in  which  the  dimension 

was  takejij  whether  inches,  feet,  or  rods,  etc. 

EXAMPLES 

1.  A  certain  garden,  in  form  of  a  long  square^  is 
96  feet  long,  and  54  feet  wide ;  how  many  square 
feet  of  ground  are  contained  in  it  ? 

Ans.  96X54=5184  square  feet. 

2.  A  lot  of  land,  in  form  of  a  long  square,  is 
120  rods  in  length,  and  60  rods  wide;  how  many 
acres  are  in  it?  120x60=7200  sq.  rods.  And 
7200-^-160=45  acres,  A7is. 

Note. — The  learner  must  recollect  that  feet  in 
length,  multipled  by  feet  in  breadth,  produce  square 
feet;  and  the  same  of  the  other  denominations  of 
lineal  measure. 

Note. — Both  the  length  and  breadth,  if  not  in 
units  of  the  same  denomination,  must  be  made  so 
before  multiplying. 

3.  How  many  acres  are  in  a  field  of  oblong 
form,  whose  length  is  14,5  chains,  and  breath  9,75 
chains?  Ans.  14ac.  Orood,  22rods. 

Note. — The  Gunter's  chain  is  66  feet,  or  4  rods, 
long,  and  contains  100  links.  Therefore  if  dimen- 
sions be  given  in  chains  and  decimals,  point  off 
from  the  product  one  more  decimal  place  than  are 


MENSURATION  OR  PRACTICAL  GEOMETRY.    145 

contained  m  both  factors,  and  it  will  be  acres  and 
decimals  of  an  acre ;  if  in  chains  and  links,  do  the 
same,  because  links  are  hundredths  of  chains,  and 
therefore  the  same  as  decimals  of  them.  Or,  as  1 
chain  wide,  and  10  chains  long,  or  10  square  chains, 
or  100000  square  links,  make  an  acre,  it  is  the  same 
as  if  you  divide  the  links  in  the  area  by  100000. 

4.  If  a  board  be  21  feet  long  and  18  inches 
broad,  how  many  square  feet  are  contained  in  it? 

18  inches=l,5  foot;  and  21X1,5=31,5  ft.,  Ans. 

Or,  in  measuring  boards,  you  may  multiply  the 
length  in  feet  by  the  breadth  in  inches,  and  divide 
the  product  by  12 ;  the  quotient  will  give  the  an- 
swer in  square  feet,  etc.  21x18 

Thus,  in  the  last  example,  =31Jsq.  ft., 

as  before.  12 

5.  If  a  board  be  8  inches  wide,  how  much  in 
length  will  make  a  foot  square  ? 

Rule. — Divide  144  by  the  width;  thus^  8)144 

Ans.  18  in, 

6.  If  a  piece  of  land  be  5  rods  wide,  how  many 
rods  in  length  will  make  an  acre  ? 

Rule. — Divide  160  by  the  width,  and  the  quo- 
tient  will  be  the  length  required;  thus, 
5)160 

Ans,  32  rods  in  length. 


146       orton's  lightning  calculator. 

Note. — When  a  board,  or  any  otter  surface,  is 
wider  at  one  end  than  tlie  other,  but  yet  is  of  a 
true  taper,  you  may  take  the  breadth  in  the  middle, 
or  add  the  widths  of  both  ends  together,  and  halve 
the  sum  for  the  mean  width ;  then  multiply  the 
said  mean  breadth  in  either  case  by  the  length  j 
the  product  is  the  answer  or  area  sought. 

7.  How  many  square  feet  in  a  board,  10  feel 
long  and  13  inches  wide  at  one  end,  and  9  inches 
wide  at  the  other  ?         13-|-9 

1=11  in.,  mean  width. 

2       ft.     in. 
10X11 

=r9Jft.,  Alls. 

12 

B.  IIow  many  acres  are  in  a  lot  of  land  which  is 
40  rods  long,  and  30  rods  wide  at  one  end,  and  20 
rods  wide  at  the  other? 
30+20 

=25  rods,  mean  width. 

2  Then,  25x40 

=6 J  acres,  Ans. 

160 
9    If  a  farm  lie  250  rods  on  the  road,  and  at  one 
end  be  75  rods  wide,  and  at  the  other  55  rods  wide, 
how  many  acres  does  it  contain  ? 

Ans.  101  acres,  2  roods,  10  rods. 

N.  B. — Always  arrange  your  example  as  above, 
and  cancel  the  factors  common  to  both  terms  before 
multiplying 


MENSURATION  OR  PRACTICAL  GEOMETRY.    U7 

Case  3. —  To  measure  the  surface  of  a  triangle. 

Definition. — A  triangle  is  any  three-cornered 
figure  which  is  bounded  by  three  right  lines.* 

Rule. — Multiply  the  hose  of  the  given  triangle 

into  half  its  perpendicular  hight,  or  half  the  base 

into  the  whole  perp3ndicularj  and  the  product  will 

be  the  area. 

EXAMPLES. 

1.  Required  the  area  of  a  triangle  whose  base  or 
longest  side  is  32  inches,  and  the  perpendicular 
bight  14  inches. 

14:-7-2r=:7=half  the  perpendicular.     And 
32xT=224sq.  in.,  Ans. 

2.  There  is  a  triangular  or  three-cornered  lot  of 
land  whose  base  or  longest  side  is  51 J  rods;  the 
perpendicular,  from  the  corner  opposite  to  the  base, 
measures  44  rods ;  how  many  acres  does  it  contain  ? 

44-j-2=:22=i:half  the  perpendicular. 
And  51,5X22 

=7  acres,  13  rods,  Ans. 

160 

Joists  and  planks  are  measured  by  the  folloioing: 

Rule. — Find  the  area  of  one  side  of  the  joist  or 

plank  by  one  of  the  preceding  rules ;  then  multiply 

it  by  the  thickness  in  inches^  and  the  last  product 

will  be  the  superficial  content. 

*  A  triangle  may  be  either  right-angled  or  oblique. 

N 


148  orton's  LIaIITNI^'G  calculator. 

EXAMPLES. 

1.  Wnat  is  the  area,  or  superficial  content,  or 
board  measure,  of  a  joist,  20  feet  long,  4  inches 
wide,  and  3  incLes  thick?     20x4 

X3=20ft.,  Ans. 

12 

2.  If  a  plank  be  32  feet  long,  17  inches  wide, 
and  3  inches  thick,  what  is  the  board  measure  of 
it?  ^ns.  136  feet 

Note. — There  are  some  numbers,  the  sum  of 
whose  squares  makes  a  perfect  square ;  such  are  3 
and  4,  the  sum  of  whose  squares  is  25,  the  square 
root  of  which  is  5 ;  consequently,  when  one  leg 
of  a  right-angled  triangle  is  3,  and  the  other  4, 
the  hypotenuse  must  be  5.  And  if  3,  4,  and  5,  be 
multiplied  by  any  other  numbers,  each  by  the  same, 
the  products  will  be  sides  of  true  right-angled  tri- 
angles. Multiplying  them  by  2,  gives  6,  8,  and  10, 
by  3,  gives  9,  12,  and  15 ;  by  4,  gives  12,  16,  and 
20,  etc.;  all  which  are  sides  of  right-angled  tri- 
angles. Hence  architects,  in  setting  off  the  corners 
of  buildings,  commonly  measure  6  feet  on  one  side, 
and  8  feet  on  the  other ;  then,  laying  a  10-foot  pole 
across  from  those  two  points,  it  makes  the  corner 
a  true  right-angle. 

N.  B. — The  solutions  of  the  foregoing  problems 
are  all  very  brief  by  canceling. 


MENSURATION  OR  PRACTICAL  GEOMETllY.     149 

To  Jind  iJie  area  of  any  triangle  when  the  three  sides 
only  are  given. 
Rule. — From  half  the  sum  of  the  three  sides  sub- 
tract each  side  severally;  multiply  these  three  re- 
mainders and  the  said  half  sum  continually  together  ; 
f.Iien  the  square  root  of  the  lust  product  will  he  tJie 
area  of  the  triangle, 

EXAMPLE. 
Suppose   I  have  a  triangular   fish-pond,  whose 
three  sides  measure  400,  348,  and  312yds;   what 
quantity  of  ground  does  it  cover? 

Ans.  10  acres,  3  roods,  8-f  rods. 

Note. — If  a  stick  of  timber  be  hewn  three 
square,  and  be  equal  from  end  to  end,  you  find 
the  area  of  the  base,  as  in  the  last  question,  in 
inches;  multiply  that  area  by  the  whole  length, 
and  divide  the  product  by  144,  to  obtain  the  solid 
content. 

If  a  stick  of  timber  be  hewn  three  square,  be  12 
feet  long,  and  each  side  of  the  base  10  inches,  the 
perpendicular  of  the  base  being  8f  inches,  what  is 
its  solidity?  Ans.  3,6-j-feet. 

PROBLEM  1. 

The  diameter  given,  to  find  the  circumference. 

Rule. — As  7  are  to  22,  so  is  the  given  diameter 
to  the  circumference ;  or,  more  exactly,  as  113  are 
to  355,  so  is  the  diameter  to  the  circumference,  etc. 


150        orton's  lightninq  calculator. 

EXAMPLES. 

1.  What  is  the  circumference  of  a  wheel,  whose 
diameter  is  4  feet? 

As  7  :  22  :  :  4  :  12,57+ft.,  the  cir.-am.,  Aits. 

2.  What  is  the  circumference  of  a  circle,  whoso 
diameter  is  35  rods  ? 

As  7  :  22  :  :  35  :  110  rods,  AjiS. 

Note. — To  find  the  diameter  when  the  circum- 
ference is  given,  reverse  the  foregoing  rule,  and  say, 
as  22  are  to  7,  so  is  the  given  circumference  to  the 
required  diameter;  or,  as  355  are  to  113,  so  is  the 
circumference  to  the  diameter. 

3.  What  is  the  diameter  of  a  circle,  whose  cir- 
cumference is  110  rods? 

As  22  :  7  :  :  110  :  35  rods,  the  diam.,    Ans. 

Ca.se  5. — To  find  how  many  solid  feet  a  round 
stick  of  timber^  equally  thick  from  end  to  end^ 
will  contain^  when  hewn  square. 
EuLE. — Multiply  twice  the  square  of  its  semi-dC- 
ameter^  in  inches^  by  the  length  in  thp.ftet;  then  divide 
the  product  by  144,  and  the  quotient  will  be  the  an- 
swer. 

N.  B. — When  multiplication  and  division  are 
combined,  always  cancel  like  factors.  When  the 
numbers  are  properly  arranged,  a  few  clips  with 
the  pencil,  and,  perhaps,  a  trifling  multiplication 
will  suffice. 


MENSURATION  OR  PRACTICAL  GEOMETRY.      151 

EXAMPLES. 
1.  If  the  diameter  of  a  round  stick  of  timber 
be  22  inches,  and  its  length  20  feet,  how  many  solid 
feet  will  it  contain  when  hewn  square? 
11X11X2X20 

Half  diameter=ll,  and =33,  6-|- ft. 

144 
the  solidity  when  hewn  square,  the  answer. 
Case  6. —  To  find  liow  many  feet  of  square  edged 
hoards,  of  a  given  ihicJcness,  can  be  sawn  from  a 
log  of  a  given  diameter. 

Rule. — Find  the  solid  content  of  the  log,  when 
mide  square,  hy  the  last  case;  then  say,  as  the 
thickness  of  the  board,  including  the  saw  calf,  is  to 
the  solid  feet,  so  is  12  inches  to  the  number  of  feet 
of  boards, 

EXAMPLES. 
1.  How  many  feet  of  square  edged  boards,  IJ 
inch  thick,  including  the  saw  calf,  can  be  sawn  from 
a  log  20  feet  long, and  24  inches  diameter? 
12X12X2X20 

=40ft.  solid  content  when  hewn  sq. 

144 

As  IJ-  :  40  :  :  12  :  384  feet,  Ans- 
2    How  many  feet  of  square  edged  boards,  1^ 
inch  thick,  including  the  saw  gap,  can  be  sawn  from 
a  log  12  feet  long,  and  18  inches  diameter? 

Ans.  108  feet. 


152  ORTON'S  LiaHTNINQ  CALCULATOR. 

Note. — A  short  rule  for  finding  the  number  of 
feet  of  one  inch  boards  that  a  log  will  make,  is  to 
deduct  J  of  its  diameter  in  inches,  and  J  of  its 
length  in  feet ;  then  for  each  inch  of  diameter  that 
remains,  reckon  1  board  of  the  same  width  as  this 
reduced  diameter,  and  of  the  same  length  as  this 
reduced  length  of  the  log :  thus  a  log  12  feet  long, 
and  12  inches  through,  gives  9  boards,  9  feet  long,- 
9  inches  wide,  or  60f  feet — a  log  16  feet  long,  and 
16  inches  through,  gives  12  boards,  12  inches 
wide,  12  feet  long,  or  144  feet. 

In  measuring  timber,  however,  you  may  multiply 
the  breadth  in  inches  by  the  depth  in  inches,  and 
that  product  by  the  length  in  feet ;  divide  this  last 
product  by  144  and  the  quotient  will  be  the  solid 
content  in  feet,  etc. 

How  many  solid  feet  does  a  piece  of  square  tim- 
ber, or  a  block  of  marble    contain,  if  it  be   16 
inches  broad,  11  inches  thick,  and  20  feet  long? 
16X  11X20=3520,  and  3520--144=24,4+sol.  ft. 
Case  8. —  To  find  the  solidity  of  a  cone  or  pyramid 

whether  round,  square,  or  triangular. 

Definition. — Solids  which  decrease  gradually 
from  the  base  till  they  come  to  a  point,  are  generally 
called  cones  or  pyramids,  and  are  of  various  kinds, 
according  to  the  figure  of  their  bases;  round, 
gquare,  oblong,  triangular,  etc. ;  the  point  at  the 
top  ib  called  the  vertex,  and  a  line  drawn  frcm  the 


MENSURATION  DR  PRACTICAL  GEOMETRY.    153 

vertex,   perpendicular  to   tlie  base,   is  called   the 
hight  of  the  pyramid. 

KuLE. — Find  the  area  of  the  hase^  whether  romid, 
Sijuare,  ohlong,  or  triangular ,hy  someone  oj  the  fore- 
go ng  rules,   as  the  case  may  he;  then  muLtiply  this 
area  hy  one-third  of  the  hight,  and  the  product  will 
he  the  solid  content  of  the  pyramid, 
EXAMPLES. 
1 .  What  is  the  content  of  a  true-tapered  round 
Btick  of  timber,  24  feet  perpendicular  length,  15 
inches  diameter  at  one  end,  and  a  point  at  the  other? 
15xl5x,7854x8 

=9,8175  solid  feet,  Ans. 

144 
To  find  the  solid  content  of  a  frustum  af  a  cone. 
What  is  the   solid   content  of  a  tapering  round 
stick  of  timber,   whose     greatest   diameter  is  13 
inches,  the  least  6J  inches,  and  whose  length  is  24 
feet,  calculating  it  by  both  rules  ? 

Rule.  2  — Multiply  each  diameter  into  itself;  mul- 
tiply on^  diameter  hy  the  other;  multiply  the  sum 
of  these  "products  hy  the  lengths;  annex  two  ciphers 
to  the  product,  and  divide  it  hy  382 ;  the  quotient 
will  he  the  content,  which  divide  hy  \^\  for  feet  as 
in  other  cases. 
;i3Xl3>fC6,5x6,5)+C13x6,5)X2400 

=1858,1 154- 

382 
And  1858,115H-144=12,903~i-ffc,  Am. 


154       orton's  liqhtnino  calculator. 

To  find  tbe  content  of  timber  in  a  tree,  multiply 
the  square  of  ^  of  the  circumference  at  the  middle 
of  the  tree,  in  inches,  by  twice  the  length  in  feet^ 
and  the  product  divided  by  144  will  be  the  content, 
extremely  near  the  truth.  In  oak,  an  allowance 
of  j*Q  or  y'2  must  be  made  for  the  bark,  if  on 
the  tree;  in  other  wood,  less  trees  of  irregular 
growth,  must  be  measured  in  parts. 

To  find  the  solid  content  of  a  frustum  or  segment 
of  a  globe. 

Definition. — The  frustum  of  a  globe  is  an}^  part 
cut  off  by  a  plane. 

Rule. —  To  three  times  the  square  of  the  semi-di- 
ameter of  the  hase^  add  tJie  square  of  the  hight ; 
multiply  this  sum  by  the  hight,  and  the  product  again 
by  .523G  ;  the  last  product  will  be  the  solid  content. 

EXAMPLE. 
If  the  hight  of  a  coal-pit,  at  the  chimney,  be  9 
feet,  and  the  diameter  at  the  bottom  be  24  feet, 
how  many  cords  of  wood  does  it  contain^^  allowing 
nothing  for  the  chimney? 
24-f-2=12r:^h'fdiam.  12X12X3=432.    9x9=81 


And432+81x9Xr5236 

18,886+cords,  Ans. 

128=:rsclid  feet  in  a  cord^ 


SHORT  RULES  FOR  THE  MECHANIO. 


Question. — A  stick  of  timber  is  carried  by  three 
men,  one  carries  at  the  end,  and  the  other  two  with 
a  lever.  How  far  should  the  lever  be  placed  from 
the  other  end,  that  each  man  may  carry  equally  ? 

Rule. — Divide  the  length  of  the  stick  by  4,  and 
the  quotient  is  the  answer. 

There  is  a  stick  of  timber,  30  feet  long,  to  be 
carried  by  3  men :  one  carries  at  the  end,  the  other 
two  carry  by  a  lever ;  how  far  must  the  lever  be 
placed  from  the  other  end,  that  each  may  carry 
equally  ?  Ans.  7|  feet  from  the  end. 
155 


CASK-GAUGING. 


Gauging  is  Ihe  art  of  measuring  the  capacity  of 
casks  and  vessels  of  any  form.  In  commerce,  most 
of  the  gauging  is  done  by  the  use  of  (he  diagonal 
rod,  which  gives  only  approximate  results,  but  suf- 
ficiently accurate  for  ordinary  purposes. 

Ullage  is  the  difference  between  the  actual  con- 
tents of  a  vessel  and  its  capacity,  or  that  part 
which  is  empty. 

To  measure  small  cylindrical  vessels. 

Rule. — Multiply  the  square  of  the  diameter,  in 
inches,  by  34,  and  that  by  the  height,  in  inches, 
and  point  off  four  figures ;  the  result  will  be  the 
capacity,  in  wine  gallons  and  decimals  of  a  gallon. 

For  beer  gallons  multiply  by  28  instead  of  34. 
156 


CASK-GAUlilNG.  157 

Example. — A  can  measures  15  inches  in  diame- 
ter, and  is  2  feet  2  inches  in  height.  How  many  gal- 
lons will  it  contain  ?  15x15  =  225  X  26  height  = 
5850  ;  5850  x  34  =  19.8900.     Ans.  ldj%%  galls. 

Casks  are  usually  regarded  as  the  two  equal  frus- 
tums of  a  cone,  and  are  very  accurately  gauged  by 
three  dimensions  as  follows  : — 

To  measure  a  cask  by  three  dimensions. 

1st.  Add  the  bung  and  head  diameters  in  inches, 
and  divide  by  2  for  the  mean  diameter. 

2d.  Multiply  the  square  of  the  mean  diameter  "by 
the  length  of  the  cask  in  inches. 

3d.  Multiply  the  last  product  by  .0034  for  wine 
gallons,  .0028  for  beer  gallons. 

Example, — How  many  wine  gallons  in  a  cask, 
the  bung  diameter  of  which  is  22  inches,  the  head 
diameter  20  inches,  and  the  length  32  inches  ? 

Work.— 22  +  20  =  42  h-  2  =  21  (mean  diame- 
ter) :  then  21  x  21  =  441  (square  of  mean  diame- 
ter), X  32  length  =  14112  x  -0034  =  47.9808. 
Ans. 

Note. — If  the  cask  is  not  full,  stand  it  on  the 
end,  and  multiply  by  the  height  of  the  liquid,  in- 
stead of  the  length  of  the  cask,  for  actual  contents. 

When  the  cask  is  much  bilged  or  rounded  from 
the  bung  to  the  head,  a  more  accurate  way  is  to 
gauge  by  four  dimensions,  as  follows  : — 


158         orton's  lightning  calculator. 

To  measure  a  cash  by  four  dimensions. 

1st.  Add  the  bung  and  head  diameters  in  inches, 
and  the  diameter  in  inches  between  bung  and  head. 

2d.  Divide  their  sum  by  3  for  the  mean  diameter. 

3d.  Multiply  the  square  of  the  mean  diameter 
by  the  length  of  the  cask  in  inches. 

4th.  Multiply  the  last  product  by  .0034  for  wine 
gallons,  .0028  for  beer  gallons. 

Example. — What  are  the  contents  in  gallons  of 
a  cask,  the  bung  diameter  of  which  is  24  inches, 
the  middle  diameter  20  inches,  the  head  diameter 
16  inches,  and  its  length  40  inches? 

Work.— 24  +  20  +  16  =  60  -f-  3  =  20  (mean 
diameter),  then  20  X  20  =  400  (square  of  mean  dia- 
meter) X  40 length  =  16000  X  .0034  =  54. 4 gallons. 

1.  The  ale  gallon  contains  282  cubic  inches. 

2.  The  wine  gallon  contains  231  cubic  inches. 
3    The  bushel  contains  2150.4  cubic  inches. 

4.  A  cubic  foot  of  pure  water  weighs  1000 
ounces  =  62^  pounds  avoirdupois. 

5.  To  find  what  weight  of  water  may  be  put  into 
a  given  vessel. 

Multiply  the  cubic  feet  by  1000 /or  the  ounces, 
or  by  62J /or  the  pounds  avoirdupois. 

6.  What  weight  of  water  can  be  put  into  a  cis- 
tern YJ  feet  square  ?     Ans.  26,361  lbs.  3  oz. 


MENSURATION  OF  PRACTICAL  GEOMETRY.  159 

To  find  the  contents  of  a  round  vessel,  wider  at 
one  end  than  th**  other. 

Rule, — Multiply  the  great  diameter  hy  the  less; 
to  this  product  add  J  of  the  square  of  their  differ- 
ence, then  multiply  hy  the  hight,  and  divide  as  in 
the  last  rule. 

Having  the  diameter  of  a  circle  given,  to  find 
the  area. 

Rule. — Multiply  half  the  diameter  hy  half  the 
circumference,  and  the  product  is  the  area ;  or, 
which  is  the  same  thing,  multiply  the  square  of  the 
diameter  hy  .7854,  and  the  product  is  the  area. 

To  find  the  solidity  of  a  sphere  or  globe. 

Rule. — Multiply  the  cuhe  of  the  diameter  by 
.5236. 

To  find  the  convex  surfiice  of  a  sphere  or  globe. 

Rule. — Multiply  iis  diameter  hy  its  circumfer- 
ence. 

To  find  the  solidity  of  a  prism. 

Rule — Multiply  the  area  of  the  hase,  or  end,  hy 
the  hight. 

How  many  wine  gallons  will  a  cubical  box  con- 
tain, that  is  10  long,  5  feet  wide,  and  4  feet  high? 

Rule. —  Take  the  dimensions  in  inches;  then  mid- 
tiply  the  length,   hreadth,  and  hight  together;  di- 
vide the  product  hy  282  for  ale  gallons,  231  for  idnc 
gallons,  and  2150  for  bushels. 
0 


160 


ORTON'S  LIGHTNING  CALCULATOR. 


In  estimating  the  capacity  of  cisterns,  resep- 
VoIps,  &c.,  the  following  table  is  used  : — 
31^  gallons  .         .         .1  barrel. 
63         "       .         .         .1  hogshead. 
Note. — The  barrels  used  in  commerce  vary  from 
30  to  45  gallons,  and  the  hogshead  from  40  to  60 
gallons,  &c. 

Hogshead  of  claret        .     46  gallons. 

"         of  brandy  55  to  60 
Puncheon  of  brandy  1 1 0  to  1 20       " 
"  rum.    100  to  110       " 

Pipe  of  port  .        .     115       " 


"      "   Madeira   . 

92 

"      "  Teneriffe  . 

.     100 

Butt  of  sherry 

.     108 

"     "   Malaga     . 

.     105 

Physicians  and  druggists,  in  compounding  medi- 
cines, divide  the  gallon  diflferently,  and  according 
to  the  following  : — 

APOTHECAKIES'  FLUID  MEASURE. 

Minima.  Drachms.  Ounces. 


60  minims  (a  drop) 
8  drachms  1  ounce 
16  ounces  1  pint 
8  pints  1  gallon   . 
Note. — The  gallon  of  the  above  measure  is  the 
same  capacity  as  the  gallon  of  liquid  measure. 


1  fluidrachm. 
.     480 

.    9600  =  128 
16800=1024  =  128 


MEASURES  OF  CAPACITY.  161 

TROY  WEIGHT. 


Grains.        Dwt. 

480 
5760  =  240 


24  grains  1  dwt. 

20  dwt.  1  ounce. 

12  ounces  1  pound. 

Note. — Troy  weight  contains  5760  grains  to  the 
pound.  Therefore,  it  will  be  seen  that  avoirdupois 
contains  1240  more  grains  to  the  pound  than  Troy 
weight. 

Apothecapjes' weight  is  used  by  physicians  and 
druggists  in  dispensing  medicines. 

APOTHECARIES'  WEIGHT. 

20  grains  1  scruple. 
3  scruples  1  drachm. 
8  drachms  1  ounce. 
12  ounces  1  pound. 
Note. — The  pound,  ounce,  and   grain  of  this 
weight  is  the  same  as  that  of  Troy  weight. 

SHOEMAKERS'  MEASURE. 
Number  one,  children's  measure,  is  4|  inches, 
and  that  every  additional  number  calls  for  an  in- 
crease of  J  of  an  inch  in  length.  Number  one 
adults'  measure  is  8J  inches  long,  with  a  gradual 
increase  of  J  of  au  inch  for  additional  numbers,  so 
that,  for  example,  number  ten  measures  11 J  inches. 
This  measure  corresponds  to  the  number  of  the 
last,  and  not  to  the  length  of  the  sole. 


Grains.    Scruples.  Drachms. 

60 
480  =    94 
5760  =  218  =  96 


BRICK  BUILDING. 


A  perch  of  stone  is  24. 15  cubic  feet ;  when  built 
in  the  wall,  22  cubic  feet  make  1  perch,  2  j  cubic 
feet  being  allowed  for  the  mortar  and  filling. 

Three  pecks  of  lime  and  four  bushels  of  sand  to 
a  perch  of  wall. 

To  find  the  number  of  perches  of  stone  in  walls. 

Rule. — Multiply  the  length  in  feet  by  the  height 
in  feet,  and  that  by  the  thickness  in  feet,  and  divide 
the  product  by  22. 

Example. — How  many  perches  of  stone  con- 
tained in  a  wall  40  feet  long,  20  feet  high,  and  18 
inches  thick  ? 

Solution. — 40  feet  length  X  20  feet  height  x  IJ 
feet«thick  =  1200  -j-  22  =  54.54  perches.     Ans. 
162 


bricklayer's  work.  163 

Note. — To  find  the  perches  of  masonry,  divide 
the  cubic  feet  by  24.75,  instead  of  22. 

Brick-work. 

The  dimensions  of  common  bricks  are  from  Tf 
to  8  inches  long,  by  4;^  wide,  and  2J  thick.  Front 
bricks  are  8;J^  inches  long,  by  4jwide,  and  2J  thick. 

The  usual  size  of  fire-bricks  is  9^  inches  long,  by 
4|  wide,  by  2|  thick. 

Twenty  common  bricks  to  a  cubic  foot  when  laid ; 
15  common  bricks  to  a  foot  of  8-inch  wall  when  laid. 

To  find  the  number  of  common  bricks  in  a  wall. 

Rule. — Multiply  the  length  of  the  wall  in  feet 
by  the  height  in  feet,  and  that  by  its  thickness  in 
feet,  and  that  again  by  20. 

Example. — IIow  many  common  bricks  in  a  wall 
40  feet  long  by  20  feet  high,  and  12  inches  thick? 

Solution. — 40  feet  length  x  20  f.et  height,  x  1 
foot  thick,  X  20  =  16000.     Ans. 

Note. — For  walls  8  inches  thick,  multiply  the 
length  in  feet  by  the  height  in  feet,  and  that  by  15. 

When  the  wall  is  perforated  by  doors  and  win- 
dows, deduct  the  sum  of  their  cubic  feet  from  the 
cubic  contents  of  the  wall,  including  the  openings, 
before  multiplying  by  20  or  15  as  before. 

Laths. 

Laths  are  1 J  to  1^  inch  wide,  by  4  feet  long,  are 
usually  set  J  inch  apart,  and  a  bundle  contains  100. 
o* 


164         orton's  lightning  calculatoe. 

IV.     OF  bricklayers'  work. 
The  principal  is  tiling,  slating,  walling  and  chim- 
ney work. 

1.   Of  Tiling  or  Slating, 

Tiling  and  slating  are  measured  hy  the  square 
of  100  feet,  as  flooring,  partitioning  and  roofing 
were  in  the  Carpenters'  work ;  so  that  there  is  not 
much  difi'erence  between  the  roofing  and  tiling; 
yet  the  tiling  will  be  the  most ;  for  the  bricklayers 
sometimes  will  require  to  have  double  measure  for 
hips  and  valleys. 

When  gutters  are  allowed  double  measure,  the 
way  is  to  measure  the  length  along  the  ridge-tile, 
and  add  it  to  the  content  of  the  roof:  this  makea 
an  allowance  of  one  foot  in  breadth,  the  whole 
length  of  the  hips  or  valleys.  It  is  usual  also  to 
allow  double  measure  at  the  eaves,  so  much  as  the 
projector  is  over  the  plate,  which  is  commonly 
about  18  or  20  inches. 

Sky-lights  and  chimney  shafts  are  generally  de- 
ducted, if  they  be  large,  otherwise  not. 

Example  1.  There  is  a  roof  covered  with  tiles, 
whose  depth  on  both  sides  (with  the  usual  allow- 
ance at  the  eaves)  is  37  feet  3  inches,  and  the 
length  45  feet ;  how  many  squares  of  tiling  are 
contained  therein  ? 


bricklayers'  work. 


165 


BY   DUODE 

CIMALS, 

FEET. 

INCHg^i 

37 

3 

45 

0 

185 

148 

11 

3 

BY   DECIMALS. 

37.25 
45 

18625 
14900 


16  76.25 


16  76     3 

2.   Of  Walhng. 

Bricklayers  commonly  measure  tlieir  work  by 
iLe  rod  of  16 J  feet,  or  272 J  square  feet.  In  some 
places  it  is  a  custom  to  allow  18  foet  to  the  rod  : 
that  is,  324  square  feet.  Sometimes  the  work  is 
measured  by  the  rod  of  21  feet  long  and  3  feer 
high,  that  is,  63  square  feet ;  and  then  no  regard 
is  paid  to  the  thickness  of  the  wall  in  measuring* 
but  the  price  is  regulated  according  to  the  thick- 
ness. 

When  you  measure  a,  piece  of  brick -work,  tho 
first  thing  is  to  inquire  by  which  of  these  ways  it 
must  be  measured ;  then,  having  multiplied  the 
length  and  breaxith  in  feet  together,  divide  the  pro- 
duct by  the  proper  divisor,  viz.:  272.25,  324  or  63, 
according  to  the  measure  of  the  rod,  and  the  quo- 
tient will  be  the  answer  in  square  rods  cf  that 
measure. 

But,  commonly,  brick  walls  that  are  measured 
by  the  rod  are  to  be  reduced  to  a  standard  thick- 


166         orton's  lighting  calculator. 

ness  of  a  brick  and  a-half,  wliicli  may  be  done  by 
the  following 

Rule. — Multiply  the  number  of  snperjicwLl  feet 
that  are  contained  in  the  wall  hy  the  numher  of 
half  bricks  which  that  wall  is  in  thickness;  one- 
third  part  of  that  product  will  be  the  content  in 
feet. 

The  dimensions  of  a  building  are  generally 
taken  by  measuring  half  round  the  outside  and 
half  round  the  inside,  for  the  whole  length  of  the 
wall ;  this  length,  being  multiplied  by  the  hight, 
gives  the  superficies.  And  to  reduce  it  to  the 
standard  thickness,  etc.,  proceed  as  above.  All  the 
vacuities,  such  as  doors,  windows,  window  backs, 
etc.,  must  be  deducted. 

To  measure  any  arched  way,  arched  window  or 
door,  etc.,  take  the  hight  of  the  window  or  dooi 
from  the  crown  or  middle  of  the  arch  to  the  bot- 
tom or  sill,  and  likewise  from  the  bottom  or  sill  to 
the  spring  of  the  arch ;  that  is,  where  the  arch 
begins  to  turn.  Then  to  the  latter  hight  add  twice 
the  former,  and  multiply  the  sum  by  the  width  of 
the  window,  door,  etc.,  and  one-third  of  the  pro- 
duct will  be  the  area,  sufficiently  near  for  practice. 

Example  1.  If  a  wall  be  72  feet  6  inches  long, 
and  19  feet  3  inches  high,  and  5 J  bricks  thick, 
how  many  rods  of  brick  work  are  contained  therein, 
when  reduced  to  the  standard  ? 


GLAZIERS    WORK. 


167 


VII.    GLAZIERS    WORK. 

Glaziers  take  their  dimensions  ir.  feet,  inches 
and  eights  or  tenths,  or  else  in  feet  and  hundredth 
parts  of  a  foot,  and  estimate  their  work  by  the 
square  foot. 

Windows  are  sometimes  measured  by  taking  the 
dimensions  of  one  pane,  and  multiplying  its  super- 
ficies by  the  number  of  panes.  But,  more  gen- 
erally, they  measure  the  length  and  breadth  of  the 
window  over  all  the  panes  and  their  frames  for  the 
length  and  breadth  of  the  glazing. 

Circular  or  oval  windows,  as  fan  lights,  etc.,  are 
measured  as  if  they  were  square,  taking  for  their 
dimensions  the  greatest  length  and  breadth,  as  a 
compensation  for  the  waste  of  glass  and  labor  in 
cutting  it  to  the  necessary  forms. 

Example  1.  If  a  pane  of  glass  be  4  feet  8| 
inches  long,  and  1  foot  4J-  inches  broad,  how  many 
(ect  of  glass  are  in  that  pane  ? 

BY  DECIMALS. 

4.729 
1.354 


BY  DUODECIMALS. 

FT. 

IN. 

p. 

4 

8 

9 

1 

4 

3 

4    8     9 

1     6  11 

0 

1     2 

2     3 

6    4  10     2    3 


18916 
23645 
14187 
4729 

6.403066 
Am.  6  feet  4  inches. 


168       orton's  lightning  calculator. 

VIII.      PLUMBER8    WORK. 

Plumbers'  work  is  generally  rated  at  so  much 
per  pound,  or  by  tlie  hundred  weight  of  112 
pounds,  and  the  price  is  regulated  according  to  the 
value  of  lead  at  the  time  when  the  work  is  per- 
formed. 

Sheet  lead,  used  in  roofing,  guttering,  etc., 
weighs  from  G  to  12  pounds  per  square  foot,  ac- 
cording to  the  thickness,  and  leaden  pipe  varies  in 
weight  per  yard,  according  to  the  diameter  of  its 
bore  in  inches. 

The  following  table  shows  the  weight  of  a  square 
foot  of  sheet  lead,  according  to  its  thickness,  reck- 
oned in  parts  of  an  inch,  and  the  common  weight 
of  a  yard  of  leaden  pipe  corresponding  to  the 
diameter  of  its  bore  in  inches: 


Thickness 
of  Lead. 

Pounds  to  a 
Square  foot. 

Bore  of 
Leaden  Pipe. 

Pounds 
per  yard. 

sV 

5.899 

f 

10 

J 

6.554 

1 

12 

HOC 

7.373 

H 

16 

>, 

8.427 

H 

18 

A 

9.831 

If 

21 

_L    _ 

11.797 

2 

21          i 

MASON  S  WORK. 


169 


Example  1.  A  piece  of  sheet  lead  measures  16 
Feet  9  iuclics  in  length,  and  6  feet  6  inches  in 
breadth;  what  is  its  weight  at  8^  pounds  to  a 
square  foot  ? 


BY  DUODECIMALS 


FEET. 

16 

6 

INCHES. 
9 
6 

100 

8 

6 
4 

0 

108 

10 

6 

BY  DECIMALS 
FEET. 

16.75 
6.5 


8375 
10050 

108.875  ftet. 


Then    1    foot    :    8J    pounds  :  :  108.875    feet 
898.21875  poundsi=8  cwt.  2 J  pounds  nearly. 


IX.      MASON  S    WORK 


Masons  measure  their  work  sometimes  by  the 
foot  solid,  sometimes  by  the  foot  superficial,  and 
sometimes  by  the  foot  in  length.  In  taking 
dimensions  they  girt  all  their  moldings  as 
joiners  do. 

The  solids  consist  of  blocks  of  marble,  stwnc 
pillars,  columns,  etc.  The  superficies  are  pave- 
ments, slabs,  chimney-pieces,  etc. 


170       obton's  lightning  calculator. 

V.    PLASTERERS     WORK. 

Plasterers'  work  is  principally  of  two  kinds; 
namely,  plastering  upon  laths,  called  ceiling^  and 
plastering  upon  walls  or  partitions  made  of  framed 
timber,  called  rendering. 

In  plastering  upon  walls,  no  deductions  are  mado 
except  for  doors  and  windows,  because  cornice?, 
festoons,  enriched  moldings,  etc.,  are  put  on  after 
the  room  is  plastered. 

In  plastering  timber  partitions,  in  large  ware- 
houses, etc.,  where  several  of  the  braces  and  larger 
timbers  project  from  the  plastering,  a  fifth  part  is 
commonly  deducted.  Plastering  between  their 
timbers  is  generally  called  rendering  between 
quarters. 

Whitening  and  coloring  are  measured  in  the 
same  manner  as  plastering ;  and  in  timbered  par- 
titions, one-fourth,  or  one-fifth  of  the  whole  area  is 
commonly  added,  for  the  trouble  of  coloring  the 
sides  of  the  quarters  and  braces. 

Plasterers'  work  is  measured  by  the  yard  square, 
consisting  of  nine  square  feet.  In  arches,  the  girt 
round  them,  multiplied  by  the  length,  will  give  the 
superficies. 

Example  1. — If  a  ceiling  be  59  feet  6  inches 
long,  and  24  feet  6  inches  broad ;  how  many  yards 
does  that  ceiling  contain  ? 


CISTERNS.  17J 

PROBLEM  L 

To  find  tlie  solid  content  of  a  Dome,  having  tJic 
hight  and  the  dimensions  of  its  base  given, 

lluLE. — Multiply  the  area  of  the  base  by  the 
hightj  and  f  of  the  product  will  be  the  solidity. 

Example  1. — What  is  the  solidity  of  a  dome,  in 
the  form  of  a  hemisphere,  the  diameter  of  the  cir- 
cular base  being  60  feet  ? 

GO'X. 7854=2827.44  area  of  the  base. 

Then  f  (2827.44x30)=56548.8  cubic  feet. 
Am. 

PROBLEM  IL 

To  find  the  superficies  of  a  dome,  having  the  hight 
and  dimensions  of  its  base  given. 

Rule. — Multiply  the  area  of  the  base  by  2,  and 
the  product  will  be  the  superficial  content  required  ; 
or,  multiply  the  sq^aare  of  the  diameter  of  the  base 
by  1.5708. 

For  an  Elliptical  Dome. — Multiply  the  two 
diameters  of  the  base  together,  and  that  product  by 
1.5708,  the  last  product  will  be  the  area,  sufficiently 
correct  for  practical  purposes. 

XI.   CISTERNS. 

Cisterns  are  large  reservoirs  constructed  to  hold 
water,  and  to  be  permanent,  should  be  made  either 
of  brick  or  masonry. 
P 


172       orton's  lightnin*!  calculator. 

It  frequently  occurs  that  they  are  to  be  so  con- 
structed as  to  hold  given  quantities  of  water,  and 
it  then  becomes  a  useful  and  practical  problem  to 
calculate  their  exact  dimensions. 

How  do  you  find  the  number  of  hogsheads 
which  a  cistern  of  given  dimensions  will  contain  ? 

1st.  Find  the  solid  content  of  the  cistern  in 
cubic  inches. 

2d.  Divide  the  content  so  found  by  14553,  and 
the  quotient  will  be  the  number  of  hogsheads. 

If  the  hight  of  a  cistern  be  given,  how  do  you 
find  the  diameter,  so  that  the  cistern  shall  con- 
tain a  given  number  of  hogsheads  ? 

1st.  Reduce  the  hight  of  the  cistern  to  inches, 
and  the  content  to  cubic  inches. 

2d.  Multiply  the  hight  by  the  decimal  .7854. 

2.  Divide  the  content  by  the  last  result,  and 
extract  the  square  root  of  the  quotient,  which  will 
be  the  diameter  of  the  cistern  in  inches. 

EXAMPLE. 

If  the  diameter  of  a  cistern  be  given,  how  do  yon 
find  the  hight,  so  that  the  cistern  shall  contain  a 
given  number  of  hogsheads  ? 

1st.  Reduce  the  content  to  cubic  inches. 

2d.  Reduce  the  diameter  to  inches,  and  then  mul- 
tiply its  square  by  the  decimal  .7854. 


MEASURING  GRAIN. 


By  the  United  States  standard,  2150  cubic  inches 
make  a  bushel.  Now,  as  a  cubic  foot  contains 
1Y28  cubic  inches,  a  bushel  is  to  a  cubic  foot  as 
2150  to  1Y28;  or,  for  practical  purposes,  as  4  to 
5.  Therefore,  to  convert  cubic  feet  to  bushels,  it 
is  necessary  only  to  multiply  by  |. 

To  measure  the  bushels  of  grain  in  a  granary/. 

Rule. — Multiply  the  length  in  feet  by  the 
breadth  in  feet,  and  that  again  by  the  depth  in 
feet,  and  that  again  by  J.  The  last  product  will 
l>e  the  number  of  bushels  the  granary  contains. 

Example. — How  many  bushels  in  a  bin  10  feet 
long,  4  feet  wide,  and  4  feet  deep. 

Work. — 10  feet  length  X  4  feet  breadth  x  4  feet 
depth  =  160  cubic  feet ;  then  160  x  |  =  128.  Ans. 
It3 


174        orton's  lightning  calculator. 

3d.  Divide  the  content  by  tlie  last  result,  and 
the  quotient  will  be  the  hight  in  inches. 

XII.   BINS  FOR  GRAIN. 

Having  any  number  of  bushels,  how  then  will 
you  find  the  corresponding  number  of  cubic  feet  ? 

Increase  the  number  of  bushels  one -fourth 
itself,  and  the  result  will  be  the  number  of  cubic 
feet. 

How  will  you  find  the  number  of  bushels  which 
a  bin  of  a  given  size  will  hold  ? 

Find  the  content  of  the  bin  in  cubic  feet ;  then 
diminish  the  content  by  one-fifth,  and  the  resuU 
will  be  the  content  in  bushels. 

How  will  you  find  the  dimensions  of  a  bin  which 
shall  contain  a  given  number  of  bushels  ? 

Increase  the  number  of  bushels  one -fourth 
itself,  and  the  result  will  show  the  number  of  cubic 
feet  which  the  bin  will  contain.  Then,  when  two 
dimensions  of  the  bin  are  known,  divide  the  last 
result  by  their  product,  and  the  quotient  will  be  the 
other  dimension. 


WEIGHTS  AND  MEASURES. 


176 


From 


To 


OQ 


Jan.... 
Feb.... 
March 
April .. 
May... 
June... 
July... 
Aug ... 
Sept ... 
Oct.... 
Nov.... 
Dec... 


365 

334 

306 

275 

245 

214 

184 

153 

122 

92 

61 

31 


31 
365 
337 
306 
276 
245 
215 
184 
153 
123 
92 
62 


59 
28 
365 
334 
304 
273 
243 
212 
181 
151 
120 
90 


90 
59 
31 
365 
335 
304 
274 
243 
212 
182 
151 
121 


120 

89 

61 

30 

365 

334 

304 

273 

242 

212 

181 

151 


151 

120 

92 

61 

31 

365 

335 

304 

273 

243 

212 

182 


181 

150 

122 

91 

61 

80 

365 

334 

303 

273 

242 

212 


212 

181 

153 

122 

92 

61 

31 

365 

334 

304 

273 

243 


243 

212 

184 

153 

123 

92 

62 

31 

365 

335 

304 

274 


273 

242 

214 

183 

153 

122 

92 

61 

30 

365 

334 

304 


304 

273 

245 

214 

184 

153 

123 

92 

61 

31 

365 

335 


331 

303 

275 

244 

214 

183 

153 

122 

91 

61 

30 

365 


TABLE  SHOWINa  DIFFERENCE  OF  TIME  AT  12 

o'clock  (noon)  at  new  YORK. 

Boston 12.12  p.  m. 

Quebec 12.12    " 

Portland 12.15    " 

London 4.55    " 

Paris 5.05    " 

Rome 5.45    " 

Constantinople  6.41    " 

Vienna 6  00    « 

St.  Petersburg..  6.57    " 

Pekin,    night...  12.40  a.  m. 


New  York 12.00  n. 

Buffalo 11.40  A.  M. 

Cincinnati 11.18    " 

Chicago 11.07    " 

St.  Louis 10.55    " 

San  Francisco...     8.45    " 

New  Orleans 10.56    " 

Washington 11.48    " 

Charleston 11.36    " 

Havana 11.25    " 


TROT  WEIGHT. 

By  this  weight  gold,  silver,  platina  and  precious 

stones,  except  diamonds,  are  estimated. 

20  Mites 1  Grain.  I  20  P^jnnywts 1  Ounce. 

20  Grains....  1  Pennywt.     j  12  Ounces 1  Pound. 

Any  quantity  of  gold  i»  supposed  to  be  divided 
p* 


176        orton's  lightning  calculator. 

into  24  parts,  called  carats.  If  pure,  it  is  said  to 
be  24  carats  fine;  if  there  be  22  parts  of  pure  gold 
and  2  parts  of  alloy,  it  is  said  to  be  22  carats  fine 
The  standard  of  American  coin  is  nine-tenths  pure 
gold,  and  is  worth  $20.67.  What  is  called  the 
new  standard  J  used  for  watch  cases,  etc.,  is  18  carats 
fine.  The  term  carat  is  also  applied  to  a  weight  of 
3J  grains  troy,  used  in  weighing  diamonds ;  it  is 
divided  into  4  parts,  called  grains  ;  4  grains  troy 
are  thus  equal  to  5  grains  diamond  weight. 

apothecaries'  weight USED    IN   MEDICAL   PEESCRIPTIONS. 

The   pound   and   ounce  of  this  weight  are  the 

same  as  the  pound  and  ounce  troy,  but  diflferently 

divided. 

20  Grains  Troy...  1  Scruple.  I    8  Drachms...!  Ounce  Troy. 
8  Scruples 1  Drachm.  |  12  Ounces....l  Pound  Troy. 

Druggists  huy  their  goods  by  avoirdupois  weight. 

AVOIRDUPOIS   WEIGHT. 

By  this  weight  all  goods  are  sold  except  those 
named  under  troy  weight. 

27|J  Grains 1  Dram. 

16     Drams 1  Ounce. 

16     Ounces 1  Pound. 

28  Pounds 1  Quarter. 

4  Quarters  or  100  pounds 1  Hundred  Weight. 

20  Hundredweight 1  Ton. 

The  grain  avoirdupois,  though  never  used,  is  the 
same  as  the  grain  in  troy  weight.  7,000  grains 
make  the  avoirdupois  pound,  and  5,760  grains  the 


WEIGHTS  AND  MEASURES.  177 

troj  pound.  Therefore,  tlie  troy  pound  is  loss  iLan 
the  avoirdupoi3  pound  in  the  proportion  of  14  to 
17,  nearly;  but  the  troy  ounce  is  greater  than  the 
avoirdupois  ounce  in  the  proportion  of  79  to  72, 
nearly.  In  times  past  it  was  the  custom  to  allow 
112  pounds  for  a  hundred  weight,  but  usage,  as 
well  as  the  laws  of  a  majority  of  the  States,  at  tho 
present  time  call  100  pounds  a  hundred  weight. 

apothecaries'  fluid  measurk. 

60  Minims 1  Fluid  Drachm. 

8  Fluid  Drachms 1  Ounce  (Troy). 

16  Ounces  (Troy) 1  Pint. 

8  Pints 1  Gallon. 

MEASURE  OP  CAPACITY  FOR  ALL  LIQUIDS. 

6  Ounces  Avoirdupois  of  water  make  1  Gill. 

4  Gills 1  Pint       =  34|  Cubic  Inches  (nearly). 

2  Pints 1  Quart    =  69|  do 

4  Quarts 1  Gallon  =277J  do 

31J  Gallons 1  Barrel, 

42  Gallons 1  Tierce. 

63  Gallons,  or  2  bbls 1  Hogshead. 

2  Hogsheads IPipeorButt^ 

2  Pipes 1  Tun. 

The  gallon  must  contain  exactly  10  pounds  avoir- 
dupois, of  pure  water,  at  a  temperature  of  62°, 
the  barometer  being  at  30  inches.  It  is  the 
standard  unit  of  measure  of  capacity  for  liquids 
and  dry  goods  of  every  description,  and  is  ^  larger 
than  the  old  wine  measure,  -j^^  larger  than  the  old 


178  ORTON  A  LIGHTNING  CALCULATOR. 

dry  measure,  and  5*5  less  than  the  old  ale  measure. 
The  wine  gallon  must  ccntain  231  cubic  inches. 

MEASURE  OP   CAPACITY   FOR  ALL  DRY  GOODS. 

4  Gills 1  pint       =»     34f  cubic  inclis( nearly) 

2  Pints 1  quart     =     69|  cubic  inches. 

4  Quarts 1  gallon     «=  277J  cubic  inches. 

2  Gallons 1  peck      =  654J  cubic  inches. 

4  Pecks,  or  8  gals.  1  bushel    =2150^  cubic  inches. 

8  Bushels 1  quarter  =     10^  cubic  feet  (nearly). 

When  selling  the  following  articles  a  barrel 
weighs  as  here  stated  : 

For  rice,  600  lbs.;  flour,  196  lbs.;  powder,  25 
lbs.;  corn,  as  bought  and  sold  in  Kentucky,  Ten- 
nessee, etc.,  5  bushels  of  shelled  corn — as  bought 
and  sold  at  New  Orleans,  a  flour-barrel  full  of  ears: 
potatoes,  as  sold  in  New  York,  a  barrel  contains  2J 
bushels;  pork,  a  barrel  is  200  lbs.,  distinguished 
in  quality  by  "clear,"  "mess,"  "prime;"  a  barrel 
of  beef  is  the  same  weight. 

The  legal  bushel  of  America  is  the  old  Win- 
chester measure  of  2,150.42  cubic  inches.  The 
imperial  bushel  of  England  is  2,218.142  cubic 
inches,  so  that  32  English  bushels  are  about  equal 
to  33  of  ours. 

Although  we  are  all  the  time  talking  about  the 
price  of  grain,  etc.,  by  the  bushel,  we  sell  by 
weight,  as  follows : 

Wheat,  beans,  potatoes,  and  clover- seed,  60  lbs. 


WEIGHTS  AND  MEASURES.  179 

to  tlio  bushel ;  corn,  rye,  flax-seed,  and  onions,  56 
lbs.;  corn  on  the  cob,  70  lbs.;  buckwheat,  52  lbs.; 
barley  48  lbs.;  hemp-seed,  44  lbs.;  timothy -seed, 
45  lbs.;  castor  beans,  46  lbs.;  oats,  35  lbs.;  bran, 
20  lbs.;  blue-grass  seed,  14  lbs.;  salt — the  real 
weight  of  coarse  salt  is  85  lbs.;  dried  apples,  24 
lbs.;  dried  peaches,  33  lbs.,  according  to  some 
rules,  but  others  are  22  lbs.  for  a  bushel,  while 
in  Indiana,  dried  apples  and  peaches  are  sold  by 
the  heaping  bushel;  so  are  potatoes,  turnips, 
onions,  apples,  etc.,  and  in  some  sections  oats  are 
heaped.  A  bushel  of  corn  in  the  ear  is  three  heaped 
half  bushels,  or  four  even  full. 

In  Tennessee  a  hundred  ears  of  corn  is  some- 
times counted  as  a  bushel. 

A  hoop  18J  inches  diameter,  8  inches  deep, 
holds  a  Winchester  bushel.  A  box,  12  inches 
sauare,  7  and  7^5  deep,  will  hold  half  a  bushel. 
A  heaping  bushel  is  2,815  cubic  inches. 

CLOTH  MEASURE. 

2^  Inches 1  nail. 

4  Nails 1  quarter  of  a  yard 

4  Quarters 1  yard. 

FOREIQH   CLOTH   MEASURE. 

2J  Quarters 1  Ell  Hamburgh. 

3  Quarters 1  Ell  Flemish. 

5  Quarters 1  Ell  English 

6  Quarters I  Ell  French. 


180  ORTON*S  LiaHTNlNQ  CALCULATOR. 

MEASURE  OP  LENGTH. 

12  Inches 1  foot. 

3  Feet 1  yard. 

5J  Yards 1  rod,  pole,  or  porch. 

40  Poles 1  furlong. 

8  Furlongs,  or  1,760  yds,  1  mile. 

««-AM''- j^'TyLLlT/"- 

By  scientific  persons  and  revenue  officers,  tho 
inch  is  divided  into  tenths,  hundredths,  etc.  Among 
mechanics,  the  inch  is  divided  into  eighths.  The 
division  of  the  inch  into  12  parts,  called  lines,  is 
not  now  in  use. 

A  standard  English  mile,  which  is  the  measure 
that  we  use,  is  5,280  feet  in  length,  1,760  yards,  or 
320  rods.  A  strip,  one  rod  wide  and  one  mile 
long,  is  two  acres.  By  this  it  is  easy  to  calculate 
the  quantity  of  land  taken  up  by  roads,  and  also 
how  much  is  wasted  by  fences. 

qunter's  chain. 

USED  FOB  LAND  HEASURB 

7,-^^  Inches 1  Link. 

100  Links,  or  66  feet,  or  4  poles 1  Chain. 

10  Chains  long  by  1  broad,  or  10  )    ..  . 

square  chains | 

80  Chains 1  Mile. 


WEIGHTS  AND  MEASURES.  181 

SURFACE   MEASURE. 

144  Sq.  inches  1  sq.  foot  [    40  Sq.  perches  I  rood 

9  Sq.  feet     1  sq.  yard  |      4  Roods 1  acre 

80J  Sq.  yards  1  sq.rdorprch  ^  640  Acres  1  sq.  mile 

Measure  209  feet  on  each  side,  and  you  have  a 
square  acre,  within  an  inch. 

The  following  gives  the  comparative  size,  in 
square  yards,  of  acres  in  different  countries : 

English  acre,  4,840  square  yards ;  Scotch,  6,150; 
Irish,  7,840;  Hamburgh,  11,545;  Amsterdam, 
9,722;  Dantzic,  6,650;  France  (hectare),  11,960; 
Prussia  (morgen),  3,053. 

This  difference  should  be  borne  in  mind  in  read- 
ing of  the  products  per  acre  in  different  countries. 
Our  land  measure  is  that  of  England. 

GOVERXMENT  LAND  MEASURE. 

A  Township — 36  sections,  each  a  mile  square. 

A  section — 640  acres. 

A  quarter  section,  half  a  mile  square — 160  acres. 

An  eighth  section,  half  a  mile  long,  north  and 
south,  and  a  quarter  of  a  mile  wide — 80  acres. 

A  sixteenth  section,  a  quarter  of  a  mile  square — 
40  acres. 


132 


ORTON  S  LIGHTNING  CALCULATOE,. 


The  sections  are  all    numbered   1  to  36,  com- 
mencing at  the  north-east  corner,  thus  : 


6 

5 

4 

3 

2 

mr.NE 

7 

8 

9 

10 

11 

12 

18 

17 

16* 

15 

14 

13 

19 

20 

21 

22 

23 

24 

30 

29 

28 

27 

26 

25 

31 

32 

33 

34 

35 

36 

The  sections  are  all  divided  in  quarters,  which 
are  named  by  the  cardinal  points,  as  in  section  1. 
The  quarters  are  divided  in  the  same  way.  The 
description  of  a  forty-acre  lot  would  read:  The 
south  half  of  the  west  half  of  the  south-west 
quarter  of  section  1  in  township  24,  north  of  range 
7  west,  or  as  the  case  might  be ;  and  sometimes 
m\\  fall  short,  and  sometimes  overrun  the  number 
of  acres  it  is  supposed  to  contain. 

SQUARE  MKASUBE — FOE  CARPENTERS,  MASONS,  ETC 

144  Sq  Inches 1  Sq  Foot. 

0  Sq  Ft,  or  1,296  Sq  In.  1  Sq  Yard. 
100  Sq  Feet 1  Sq  of  Flooring,  Roofing,  eto 

30|  Sq  Yards 1  Sq  Rod. 

86  Sq  Yards 1  Rood  of  Building. 

bcbool  section. 


WEIGHTS  AND  r.lEASURES.  183 

GEOOBAPHICAL  OR  NAtTTICAL  MEASURE. 

6  Feet 1  Fathom. 

110  Fathoms  or  660  ft.  1  Furlong. 

6075f    Feet 1  Nautical  Mile. 

3  Nautical  Miles 1  League. 

20   Leagues  or   60  |  .  j. 

Geo.  Miles...;  ^  degree, 

o.^n  Tk                          f  The  earth's  circumference 

abO  Degrees |  ^24,855^  miles,  nearly. 

The  nautical  mile  is  795|  feet  longer  than  tho 
common  mile. 

MEASURE  OF  SOLIDITY. 

1728  Cubic  Inches 1  Cubic  Foot. 

27  Cubic  Feet 1  Cubic  Yard. 

16  Cubic  Feet 1  Cord  Foot,  or  a  ft  of  wood. 

8  Cord  ft  or  128  Cubic  ft..  1  Cord. 


40  ft  of  round  or  60  ft  \ 
of  hewn  timber...  j 


ITon. 
42   Cubic  Feet 1  Ton  of  Shipping. 

ANGULAR  MEASURE,  OR  DIVISIONS  OF  THE  CIRCLE. 

GO  Seconds 1  Minute.  I  30  Degrees 1  Sign. 

60  Minutes 1  Degree.  |  90  Degrees 1  Quadrant 

360  Degrees 1  Circumference. 

MEASURE  OF  TIME. 

60  Seconds 1  Minute 

60  Minutes 1  Hour. 

24  Hours 1  Day. 

7  Days 1  Week. 

28  Days 1  Lunar  Month. 

28,  29,  30  or  31  Days 1  Cal.  Month. 

12'Cal.  Months 1  Year. 

365  Days 1  Com.  Year. 

3G6  Days 1  Leap  Year. 

365|^  Days 1  Julian  Year. 

.365  D.,  5  H.,  48  it.,  49  s 1  Solar  Year. 

365  D.,  G  n.,    9  ai.,  12  s 1  Siderial  Year. 

Q 


184        orton's  lightning  calculator. 

ROPES  AND  CABLES. 

6  Feet 1  Fathom 

120  Feet 1  Cable  Length. 

Miscellaneous  Important  Facts  about  Weights  and 
Measures, 

BOARD  MEASURE. 

Boards  are  sold  by  superficial  measure,  at  so 
much  per  foot  of  one  inch  or  less  in  thickness, 
adding  one-fourth  to  the  price  for  each  quarter- 
inch  thickness  over  an  inch. 

GRAIN  MEASURE  IN  BULK. 

Multiply  the  width  and  length  of  the  pile  to- 
gether, and  that  product  by  the  hight,  and  divide 
by  2,150,  and  you  have  the  contents  in  bushels. 

If  you  wish  the  contents  of  a  pile  of  ears  of 
corn,  or  roots,  in  heaped  bushels,  ascertain  the 
cubic  inches  and  divide  by  2,818. 

A  TON  WEIGHT. 

In  this  country  a  ton  is  2,000  pounds.  In  most 
places  a  ton  of  hay,  etc.,  is  2,240  pounds,  and  in 
some  places  that  foolish  fashion  still  prevails  of 
weighing  all  bulky  articles  sold  by  the  tun,  by  the 
"long  weight,"  or  tare  of  12  lbs.  per  cwt. 

A  tun  of  round  timber  s  40  feet ;  of  square 
timber,  54  cubic  feet. 


WEIGHTS  AND  MEASURES.  185 

A  quarter  of  corn  or  other  grain  sold  by  the 
bushel  Is  eight  imperial  bushels,  or  quarter  of  a 
tun. 

A  ton  of  liquid  measure  is  252  gallons. 

BUTTER 

Is  sold  by  avoirdupois  weight,  which  compares  with 
troy  weight  as  144  to  175 ;  the  troy  pound  being 
that  much  the  lightest.  But  175  troy  ounces 
equal  192  of  avoirdupois. 

A  firkin  of  butter  is  56  lbs.;  a  tub  of  butter  is 
84  lbs. 

THE    KILOGRAMME    OF    FRANCE 

Is  1000  grammes,  and  equal  to  2  lbs.  2  oz.  4  grs. 
avoirdupois. 

A  BALE  OF  COTTON, 

In  Egypt,  is  90  lbs.;  in  America  a  commercial  bale 
is  400  lbs.;  though  put  up  to  vary  from  280  to  720, 
in  difi"erent  localities. 

A  bale  or  bag  of  Sea  Island  cotton  is  300  lbs. 

WOOL. 

In  England,  wool  is  sold  by  the  sack  or  boll,  of 
22  stones,  which,  at  14  lbs.  the  stone,  is  308  lbs. 

A  pack  of  wool  is  17  stone,  2  lbs.,  which  is  rated 
as  a  pack  load  for  a  horse.  It  is  240  lbs.  A  tod 
of  wool  is  2  stones  of  14  lbs.  A  wey  of  wool  is  6 J 
tods.  Two  weys,  a  sack.  A  clove  of  wool  is  half 
a  stone. 


MEASUREMENT  OF  TIME. 


TIME  IS  THE  MEASURE  OF  DUEATIOS^. 


We  have  in  this  engraving  a  representation  of 
the  magnificent  transit  instrument  used  in  the 
Paris  Observatory.  It  is  made  on  the  same  model 
as  the  celebrated  one  in  the  Observatory  at  Green- 
wich. These,  and  the  one  at  the  National  Observa- 
tory at  Washington,  are  the  finest  in  the  world. 
The  instrument  is  used  for  the  purpose  of  determin- 
ing the  instant  of  time  a  heavenly  body  passes,  or 
makes  a  transit  across  the  meridian. 
186 


MEASUREMENT  OF  TIME.  18T 

From  the  time  a  star  appears  on  the  meridian, 
uutil  its  reappearance  the  next  day,  is  just  24  hours 
of  sidereal  time.  So  accurate  is  the  apparent 
movement  of  the  star,  that  it  has  never  been  known 
to  vary  the  one-hundredth  part  of  a  second  for  over 
a  thousand  years.  It  enables  us  to  correct  chro- 
nometers to  the  tenth  part  of  a  second,  so  that  sail- 
ors are  enabled  to  tell  the  precise  time  of  day,  and 
exactly  in  what  part  of  the  world  they  are. 

(Much  in  little  about  measuring  time.) 
One  of  the  first  inventions  for  measuring  time 
was  the  clepsydra,  or  water-clock,  which  was  a 
contrivance  of  the  Assyrians,  and  was  in  use  among 
them  as  early  as  the  reign  of  the  second  Sardanap- 
alus.  Clepsydra,  or  water-stealer,  it  was  called, 
from  two  words  which  have  that  meaning.  The 
instrument  was  of  various  materials ;  sometimes 
transparent,  but  generally  of  brass,  and  in  the 
shape  of  a  cylinder,  holding  several  gallons.  In 
any  case,  the  principle  on  which  it  operated  was 
the  sa  ne.  There  was  a  very  small  hole,  either  in 
the  side  or  bottom,  through  which  the  water  slowly 
trickled,  or  as  the  name  expresses  it,  stole  away, 
into  another  vessel  below.  In  the  lower  one  a  cork 
floated,  showing  the  rise  of  the  water.  By  calcu- 
lating how  many  times  a  day  the,  water  was  thus 
emptied  from  one  to  another,  they  gained  a  general 


188         orton's  lightning  calculator. 

idea  of  the  time.  The  Chinese  and  Egyptians  used 
this ;  so,  also,  did  the  Greeks  and  Romans ;  and  it 
is  stated  that  something  of  the  kind  was  found 
among  the  ancient  Britons.  It  seems  to  have  been 
one  of  the  earliest  rude  attempts,  in  many  nations, 
to  keep  a  record  of  the  hours.  The  idea  of  the 
hour-glass  must  have  grown  out  of  this.  Instead 
of  two  large  vessels,  there  were  devised  the  pear- 
shaped  glasses,  joined  by  what  may  be  called  the 
stem  ends ;  and  a  delicate  fine  sand  was  used  in- 
stead of  water.  It  was  the  invention  of  a  French 
monki 

THE  PENDULUM  AND  TELESCOPE— HOW 
INVENTED. 

In  1682,  Galileo,  then  a  youth  of  eighteen,  was 
seated  in  church,  when  the  lamps  suspended  from 
the  roof  were  replenished  by  the  sacristan,  who,  in 
doing  so,  caused  them  to  oscillate  from  side  to  side, 
as  they  had  done  hundreds  of  times  before,  when 
similarly  disturbed.  He  watched  the  lamps,  and 
thought  he  perceived  that  while  the  oscillations  were 
diminishing,  they  still  occupied  the  same  time.  The 
idea  thus  suggested  never  departed  from  his  mind  ; 
and  fifty  years  afterward  he  constructed  the  first 
pendulum,  and  thus  gave  to  the  world  one  of  the 
most  important  'instruments  for  the  measurement 
of  time.     Afterward,  when  living  at  Venice,  it  was 


MEASUREMENT  OF  TIME.  189 

reported  to  him  one  day  that  the  children  of  a  poor 
spectacle-maker,  while  playing  with  two  glasses, 
had  observed,  as  they  expressed  it,  that  things 
were  brought  nearer  by  looking  through  them  in  a 
certain  position.  Everybody  said,  How  curious  I 
but  Galileo  seized  the  idea,  and  invented  the  first 
telescope. 

THE  EQUATORIAL  TELESCOPE. 

The  Equatorial  telescope  is  an  instrument  used 
for  the  purpose  of  viewing  stars  that  appear  on  the 
horizon  in  the  east,  ascend  to  their  highest  declina- 
tion, and  descend  to  the  western  horizon.  The 
Transit  telescope  is  stationary,  and  the  star  can 
only  be  viewed  while  it  is  passing  the  disk  of  the 
instrument.  It  is  necessary  to  have  other  instru- 
ments that  will  follow  the  stars  through  the  regions 
of  the  heavens  in  which  they  are  carried  by  the 
daily  movement  of  the  earth.  By  means  of  com- 
plete machinery  the  Equatorial  accomplishes  all 
this,  carrying  the  instrument  in  one  direction  as 
fast  as  the  earth  takes  it  in  another.  The  observer 
can  thus  view  the  star  for  hours  without  changing 
his  position. 


190       orton's  lighining  calculator. 


ASTRONOMICAL  CALCULATIONS. 

A  scientific  method  of  telling  immediately  what  day 
of  the  week  any  date  transpired  or  will  transpire^ 
from  the  commencement  of  the  Christian  Era,  for 
the  term  of  three  tnousand  years. 

MONTHLY   TABLE. 

The  ratio  to  add  for  each  month  will  be  found 
in  the  following  table: 


Ratio  of  June  is 0 

Ratio  of  September  is 1 

Ratio  of  December  is 1 

Ratio  of  April  is 2 

Ratio  of  July  is 2 

Ratio  of  January  is 3 


Ratio  of  October  is 3 

Ratio  of  May  is 4 

Ratio  of  August  is 5 

Ratio  of  March  is 6 

Ratio  of  February  is 6 

Ratio  of  November  is 6 


Note. — On  Leap  Year  the  ratio  of  January  is  2,  and 
the  ratio  of  February  is  5.  The  ratio  of  the  other  tea 
months  do  not  change  on  Leap  Years. 

CENTENNIAL   TABLE. 

The  ratio  to  add  for  each  century  will  be  found 
in  the  following  table: 

Q    200,    900,  1800,  2200,  2600,  3000,  ratio  is ...0 

I    800,  1000,  ratio  is 6 

I    400,  1100,  1900,  2300,  2700,  ratio  is 5 

^    600   1200,  1600,  2000,  2400,  2800,  ratio  is 4 

3     600    1300,  ratio  is 3 

•)00,  700,  1400,  1700,  2100,  2500,  2900,  ratio  is 2 

loo!  800,  1500 ratio  is 1 


ASTRONOMICAL  CALCULATIONS.      191 

Note. — The  figure  opposite  each  century  is  iU  ratio; 
thus  the  ratio  for  200,  900,  etc.,  is  0.  To  find  the  ratio 
of  any  century,  first  find  the  century  in  tl»e  above  table, 
then  run  the  eye  along  the  line  until  you  arrive  at  tho 
end;  the  small  figure  at  the  end  is  its  ratio. 

METHOD    OP   OPERATION. 

Rule.* — To  the  given  year  add  its  fourth  part^ 
rejecting  the  fractions  ;  to  this  sum  add  the  day  of 
the  month;  then  add  the  ratio  of  the  month  and 
the  ratio  of  the  century.  Divide  this  svm  hy  7 ;  the 
remainder  is  the  day  of  the  week,  counting  Suiiday 
OS  the  first,  Monday  as  the  second,  Tuesday  as  the 
third,  Wednesday  as  the  fourth,  Thursday  as  the 
fifth,  Friday  as  the  sixth,  Saturday  as  the  seventh; 
the  remainder  for  Saturday  will  he  0  or  zero. 

Example  1. — Required  the  day  of  the  week 
for  the  4th  of  July,  1810. 

To  the  given  year,  which  is 10 

Add  its  fourth  part,  rejecting  fractions 2 

Now  add  the  day  of  the  month,  which  is 4 

Now  add  the  ratio  of  July,  which  is 2 

Now  add  the  ratio  of  1800,  which  is..,    0 

Divide  the  v^hole  sum  by  7.  7  |  18 — 4 

2 
We  have  4  for  a  remainder,  which  signifies  the 
fourth  day  of  the  week,  or  Wednesday. 

•When  dividing  the  year  by  4,  always  leave  off  tte  ccutuno!*.  We 
diviile  by  4  to  find  the  number  of  Leap  Year*. 


192       orton's  lightning  calculator. 

Note. — In  finding  the  day  of  the  week  for  the  present 
century,  no  attention  need  be  paid  ».o  the  centennial  ratio^ 
as  it  is  0. 

Example  2. — Required  the  day  of  the  week 
for  the  2d  of  June,  1805, 

To  the  given  year,  which  is 5 

Add  its  fourth  part,  rejecting  fractions 1 

Now  add  the  day  of  the  month,  which  is 2 

Now  add  the  ratio  of  June,  which  is 0 

Divide  the  whole  sum  by  7.  7  |  8—1 

T 

We  have  1  for  a  remainder,  which  signifies  the 
first  day  of  the  week,  or  Sunday. 

The  Declaration  of  American  Independence 
was  signed  July  4,  1776.  Required  the  day  of 
the  week. 

To  the  given  year,  which  is 76 

Add  its  fourth  part,  rejecting  fractions 19 

Now  add  the  day  of  the  month,  which  is 4 

Now  add  the  ratio  of  July,  which  is 2 

Now  add  the  ratio  of  1700,  which  is 2 

Divide  the  whole  sura  by  7.  7  |  103 — 5 

14 

"We  have  5  for  a  remainder,  which  signifies  the 
fifth  day  of  the  week,  or  Thursday. 

The  Pilgrim  Fathers  landed  on  Plymouth  Rock 
Dec.  20,  1620.     Required  the  day  of  the  week. 


ASTRONOMICAL  CALCULATIONS.      193 

To  tlie  given  year,  which  is 20 

Add  its  fourth  part,  rejecting  fractions 6 

Now  add  the  day  of  the  month,  which  is 20 

Now  add  the  ratio  of  December,  which  is 1 

Now  add  the  ratio  of  1600,  which  is 4 

Divide  the  whole  sum  by  7.  7  |  50- -1 

~7 

We  have  1  for  a  remainder,  wliicli  signifies  the 
first  day  of  the  week,  or  Sunday. 

On  what  day  will  happen  the  8th  of  January, 
1815?     Ans.  Sunday. 

On  what  day  will  happen  the  4th  of  May,  1810? 

On  what  day  will  happen  the  3d  of  December, 
1423?    Ans.  Friday. 

On  what  day  of  the  week  were  you  born? 

The  earth  revolves  round  the  sun  once  in  365 
days,  5  hours,  48  minutes,  48  seconds;  this  period 
is,  therefore,  a  Solar  year.  In  order  to  keep  pace 
with  the  solar  year,  in  our  reckoning,  we  make 
every  fourth  to  contain  366  days,  and  call  it  Leap 
Year.  Still  greater  accuracy  requires,  howevei, 
that  the  leap  day  be  dispensed  with  three  times 
m  every  400  years.  Hence,  every  year  (except 
the  centennial  years)  that  is  divisible  by  4  is  a 
Leap  YeaVj  and  every  centennial  year  that  ia 
divisible  by  400  is  also  a  Leap  Year.  The  next 
centennial  year  that  will  be  a  Leap  Year  h  2000. 


194       orton's  lightning  calculator. 

tor  the  practical  convenience  of  those  who  have  occasion  to 
refer  to  mensuration,  we  have  arranged  the  following  useful  -able 
of  multiples.  It  covers  the  whole  ground  of  practical  geometry; 
and  should  be  studied  carefully  by  those  who  wish  to  be  skilled  in 
this  beautiful  branch  of  mathematics : 

TABLK   or   MULTIPLES.' 

Diameter  of  a  circle  X  3-1416  —  Circumference. 
Radius  of  a  circle  X  6.283185  —  Circumference. 

Square  of  the  radius  of  a  circle  X  3.1416  —  Area. 

Square  of  the  diameter  of  a  circle  X  0-7854  —  Area. 

Square  of  the  circumference  of  a  circle  X  0.07958  ==  Area. 

Half  the  circumference  of  a  circle  X  by  half  its  diameter  =  Area. 

Circumference  of  a  circle  X  0.159155  —  Radius. 

Square  root  of  the  area  of  a  circle  X  0.56419  =-  Radius. 

Circumference  of  a  circle  X  0.31831  —  Diameter. 

Square  root  of  the  area  of  a  circle  X  112838  —  Diameter. 

Diameter  of  a  circle  X  0-86  —  Side  of  inscribed  equilateral  triangle. 

Diameter  of  a  circle  X  0.7071  —  Side  of  an  inscribed  square. 

Circumference  of  a  circle  X  0.225  —  Side  of  an  inscribed  square. 

Circumference  of  a  circle  X  0.282  —  Side  of  an  equal  square. 

Diameter  of  a  circle  X  0.88G2  —  Side  of  an  equal  square. 

Base  of  a  triangle  X  by  )4  the  altitude  —  Area. 

Multiplying  both  diameters  and  .7854  together  —  Area  of  an  ellipse. 

Surface  of  a  sphere  X  by  3^  of  its  diameter—  Solidity. 

Circumference  of  a  sphere  X  by  its  diameter  •=  Surface. 

Square  of  the  diameter  of  a  sphere  X  3.1416  —  Surface. 

Square  of  the  circumference  of  a  sphere  X  0.3183  =  Surface. 

Cube  of  the  diameter  of  a  sphere  X  0.5236  —  Solidity. 

Cube  of  the  radius  of  a  sphere  X  4-1888  —  Solidity. 

Cube  of  the  circumference  of  a  sphere  X  0.016887  —  Solidity. 

Square  root  of  the  surface  of  a  sphere  X  0.56419  —  Diameter. 

Square  root  of  the  surface  of  a  sphere  X  1-772454  —  Circumference. 

Cube  root  of  the  solidity  of  a  sph<  re  X  1-2407—  Diameter. 

Cube  root  of  the  solidity  of  a  sphere  X  3.8978  —  Circumferenco. 

Radius  of  a  sphere  X  1-1547  —  Side  of  inscribed  cube. 

Square  root  of  (>^  of  the  square  of)  the  diameter  of  a  sphsre  — 
Side  of  inscribed  cube. 

Area  of  its  base  X  by  3^  of  its  altitude  —  Solidity  of  a  cone  or  pyr- 
amid, whether  round,  squaie,  or  triangular. 

Area  of  one  of  its  sides  X  6  —  Surface  of  a  cube. 

Altitude  of  Tapezoid  X  J^  the  sum  cf  its  par-ullel  sides  —  Area 


O7(o 


